Cosmology

[YesNo]

If x divided by p leaveas a remainder of a, and x divided by q leaves a remainder of b, it is hardly surprising that x divided by ab will also leave some unique remainder as well, is it now?

That seems to be a good way to look at it with p and q prime. I didn't quite see it that way before.

[desiresjab]

That seems to be a good way to look at it with p and q prime. I didn't quite see it that way before.

Thanks. I want to unravel each function in this way. What you quoted is what I now consider the nutshell truth about the CRT. Unfortunately, that understanding alone does not allow one manipulative powers in CRT, it just clears the mind of junk associated with wondering what is going on.

Nor does going from the separate modulii to the combined one mean going backwards is necessarily easy, though I have seen it called trivial. It will take practice, like maneuvers in any enterprise require practice and familiarity. CRT and other functions of number theory have to become muscle memory. Computing backwards in the CRT will be the subject of my next post. It has to be--since I have a hunch I will at first be confused.

This is a central function of number theory, because it is certainly a central theorem of modular arithmetic, and modular arithmetic is central to number theory. People who know what they are doing use it routinely in various applications. There are encryption systems based on it. On computers, calculations involving very large numbers are routinely broken down into smaller units via the CRT and the result obtained using the separate modulii, before converting back to mod pq.

I say all this to justify my staying with the function a while longer, for unless I see plumb through it, I have not seen enough, though the effort is tedious and exhausting for me, as well. Yet, it is the way I always preferred to study math. No way school courses go slow enough for my tastes. There are too many places I want to stop for a month and investigate the way we investigate here, but that luxury is not practical, so Monday they are on to a new topic. Everything is a rush job.

We will turn the CRT inside out like a straitjacket, put it on and take it off unaided at will, as casually as we remove our daycoat upon entering the house. I ask, is there a lesser level of familiarity that would prove satisfactory with a function used constantly in multifarious applications in one's chosen field of study? Unless one can see these applications when they arise in the wild, one is tapping the cane of the blind; and unless one can successfully implement central functions in those situations, one remains in the wild.

[YesNo]

What I like to do to improve familiarity is to program the algorithms, to the extent there are algorithms.

[desiresjab]

What I like to do to improve familiarity is to program the algorithms, to the extent there are algorithms.

Yes, that is a beautiful way to get familiar. Plus, it has fringe benefits.

Time for the next phase. How much information do we need to work backwards fruitfully? Suppose we are given the naked result of the last problem. What can we do with this result?

x≡187 (mod 420)

x≡ ? (mod 3)
x≡ ? (mod 4)
x≡ ? (mod 5)
x≡ ? (mod 7). These values are easy to fill in.

x≡ 1 (mod 3)
x≡ 3 (mod 4)
x≡ 2 (mod 5)
x≡ 5 (mod 7).

Nothing difficult about that. It appears the only difficulty lies in working the CRT forward, not backwards. Backwards is, as they say, trivial.

Some good news at last!

[YesNo]

I agree that going backwards is trivial except it does require being able to factor the larger modulus into a different set of primes.

[desiresjab]

I agree that going backwards is trivial except it does require being able to factor the larger modulus into a different set of primes.

Now it is time for a simple application.

Multiply 41 times 43 via the CRT.

Of course we ignore the fact that we know this is equal to 422-1. At some point a highly observant mathematician would notice that the product he was trying to reconstruct (providing he had been given only the system below and not the two numbers to be multiplied)—he or she would notice that the two numbers were two apart anyway. This information would do him no particular good, I suppose. He would merely realize he was working out some T2-1, while the solution to the system was extracted as usual using CRT.

41≡1 (mod 5) 43≡3 (mod 5)
41≡6 (mod 7) 43≡1 (mod 7)
41≡5 (mod 9) 43≡7 (mod 9)
41≡8 (mod 11) 43≡10 (mod 11)


In typical alternate notation: (1, 3), (6, 1), (5, 7), (8, 10)

What we have to do is multiply the quantities I have bolded above.

1·3≡3 (mod 5), 6·1≡6 (mod 7), 5·7≡8 (mod 9), 8·10≡3 (mod 11). That is,

x≡3 (mod 5), x≡6 (mod 7), x≡8 (mod 9), x≡3 (mod 11).

Now all that remains is to solve the system.

3(693)-1 (mod 5)+ 6(495)-1 (mod 7)+ 8(385)-1 (mod 9)+3(315)-1 (mod 11)=

3(693)(2)+6(495)(3)+ 8(385)(4)+3(315)(8)

6(693)+18(495)+32(385)+24(315)≡1763 (mod 3465).

It is hard to believe, after what I had to go through to get this simple product, that this method could actually be shorter and easier in some context. Perhaps mark it down to my half dim understanding of byte mechanics in computers, I suppose. We see that it does work, though, and now we understand the mechanics of multiplication via the CRT.

We had to use numbers far larger than the eventual product to get our product. What is this? How could this be easier for a computer? For a really huge number in this method, I would need a commensurately huge number of terms in the addition. This is crazy. I have to be missing something. Am I?

[desiresjab]

Notice that in the last problem I chose the modulii to be used in the CRT multiplicaion. I chose 5, 7, 9, 11. None of these have any factors in common with 41 or 43 which are both prime, by design. But suppose I had chosen to multiply two composite numbers such as 88 and 63. It would have been possible to choose some of the modulii that were not relatively prime to 88 or 63 or both, and this is completely okay to do. In fact, I suspect it could be one way of shortening computer run time, since in those terms where the modulii share a factor with a multiplier 0 will be a constant out front for that term in the addition. One could perhaps tailor the modulii so that few computations were needed because most dropped off to zero due to this choice.

[desiresjab]

That wraps it up for now on the Chinese Remainder Theorem. We know how to solve a system of congruences and we know how to multiply using the CRT. I would not know where to look for more understanding. I think that will have to be encountered in the wild, where we must be ever alert for opportunities to deploy it. This goes for all the basic functions and algorithms of number theory such as

1 The Chinese Remainder Theorem
2 The Euler Phi function
3 Quadratic Reciprocity
4 The DeuceHound Ruler Function
5 Divisor and sum of divisors functions
6 Euclidian algorithm
7 Fermat's little theorem (generalized for composites as well)
8 Wilson's Theorem
9 Discrete Logarithms
10 Primite Roots
11 Modular power series
12 Properties of Congruences

This list can only grow. However, mastery of the above is a good idea for any serious student of number theory.

[desiresjab]

I have an idea for a graphically oriented computer program. Very simple--a number line stretching to the far reaches of cardinality. Marked on the line are all powers greater than 1 of every integer. As the viewer travels farther out the line at an adjustable speed, he is able to stop the progress where any two powers are relatively close together and examine the actual numbers on the line, then resume rushing outwatrd again. Quite simple. One could have a background of stars and eerie music suggestive of infinity. My own investigations (limited by thirty-two bit precision) indicate that the powers grow ever farther apart as we proceed outward, despite there being more of them the farther out we go. I do not have any sources and I do not know exactly how to approach the problem of calculating the distance between powers. Perhaps there is a rule. Some approximate formula would probably be the hope, if there was any hope, I mean to say.

[desiresjab]

Here is a short link on the subject. One is reminded of the formulas for irrationality measure, and that is even brought up in one of the posts. Of course this is only concerned with powers of 2 and 3, where my question more generally involves powers of all integers at once.

http://mathoverflow.net/questions/116840/distance-between-powers-of-2-and-powers-of-3

[YesNo]

The irrationality measure reminds me the Liouville questions we discussed earlier.

[desiresjab]

The irrationality measure reminds me the Liouville questions we discussed earlier.

That is what I meant. It reminded me of the same thing. Which is something I definitely have to get back to. But first things first. Why did I not already know that iteration of Euler's ф function eventually produces a power of 2, and from there on out power's of 2 exclusively?

I can see why powers of 2 do it, once it gets that far, but I do not immediately see why iterations of any number always devolve to a power of 2. I was just messing around with a ф calculator online. I put in a really huge number of the variety100...0001, looking for a big prime to begin with. I did not find one, but ф was usually about half the value of the iterated N, it seemed, once the function got cranking for a few iterations, and eventually it was always half. I do not have any information about how many steps it takes for this to occur. It seems like something that would have been figured out already. A little thought might reveal the answer intuitively. I hate to think too hard. What do you say about this?

[desiresjab]

Well, scratch that observation about the ф function. It was wrong. I think I am glad. Amateurs are often led astray and given to much ado about nothing, until a little more investigation reveals their haste.

[desiresjab]

I hate to keep correcting myself. I think I got sleepy and confused one investigative result with another. I do believe repeated iteration of the Euler ф function eventually yields a pure power of 2 in its chain for any whole number whatsoever, and of course from there on out, all powers of 2. It only takes one counter example.

[desiresjab]

Actually, it occurs to me now that if one proved the ф function was always even (excluding the very last iteration), that would prove the iterations must eventually descend to a power of 2. What makes me curious is that the function values seem to descend to a pure power of 2 long before they have to in the sense I just described. Many large arguments arrive at a pure power of 2 on a quite large one. 1000000, on the other hand, did not arrive until 4096.

[YesNo]

I think the phi function should be even for n > 2 based on this article: https://en.wikipedia.org/wiki/Euler's_totient_function

The function is multiplicative so we only need to consider its value for prime powers but phi of a prime power, pn, is equal to pn-1(p - 1). The p - 1 makes it even for odd prime p.

[desiresjab]

Good. I hadn't seen yet that its being a multiplicative function allowed us to ignore all but prime powers, though I usually suspect something like that in similar situations. We could just break it down as a bijection between *Z/pq and *Z/p x *Z/q, I suppose. Does a bijection make it both homomorphic and isomorphic?

[YesNo]

The bijection https://en.wikipedia.org/wiki/Bijection is a one-to-one and onto mapping between the elements of two sets. One can use a bijection to show that the two sets have the same number of elements. The only requirement is that the elements be all used once and only once between the two sets.

If one talks about various homomorphisms https://en.wikipedia.org/wiki/Homomorphism then there is an algebraic structure that the map preserves when the elements are mapped from one set to the other. That is, not any mapping between the elements of the two sets will do.

[desiresjab]

The bijection https://en.wikipedia.org/wiki/Bijection is a one-to-one and onto mapping between the elements of two sets. One can use a bijection to show that the two sets have the same number of elements. The only requirement is that the elements be all used once and only once between the two sets.

If one talks about various homomorphisms https://en.wikipedia.org/wiki/Homomorphism then there is an algebraic structure that the map preserves when the elements are mapped from one set to the other. That is, not any mapping between the elements of the two sets will do.

Trying to get my morhisms straight. The character of isomorphisms seems to change a liitle from field to field, and homomorhisms are not the same thing as homeomorhisms.

But the precise meanings of bijections, injections and surjections are super stable, from what I can tell. Finite rings, a familiar stomping ground, are surjective, bijective and injective at the same time. Who could forget that? So that matter is settled as far as finite rings are concerned.

I am looking around for something to look at. I may investigate that phi function phenomenon and see if I can find any rule to how fast an argument under iteration of the function devolves to a pure power of 2. Hopefully, the reason will be easy to spot, or it could turn into a prolonged investigation. That is not so bad either. There are always connections to be spotted between these major functions.

[YesNo]

As I see it, homeomorphisms preserve closeness rather than an algebraic structure: https://en.wikipedia.org/wiki/Homeomorphism

I suspect if one has a bijective homomorphism or a bijective homeomorphism then the inverse mapping exists. Not only is the algebra or topological structure preserved but also the number of elements involved. One should be able to go backwards having that. It has been a while since I thought about these ideas. I imagine most mappings that don't preserve anything between the two sets.

[desiresjab]

Earlier this year I went through an abstract algebra course on YouTube. I did it crazy fast--like thirty-eight lectures in three or four days. I was only able to retain some parts, but it was a good introduction to the language. They never did give a proof of quadratric reciprocity though they were hitting all around it. It amazrd me that complicated proofs and other undertakings were dealt with in a few sweeps of the chalk in that language. I guess this is because so much information is already contained and assumed in the forms they are using. One knows the engine from looking at the schematic of it rather than getting one's hands greasy down among the gears, was my impression of abstract algebra. I will probably slip in some of the language now and then. Mainly, as you know, I like greasy hands in mathematics. The reason abstract mathematics exists is because the smartest people were no longer able to see through the gears and wires of the engine to work on what had to be worked on. They knew there was structure there. If they couldn't, then I cannot either, but I am trying to be sure of where they stopped even as I try to learn the higher ways of schematics--bijections and isomorphisms et al. I want to have the same relationship with numbers they had before they made the transfer. For instance, Ramanujan took a good look at Brocard's problem. What did he see that made him decide he could not solve this one? None of us will ever see what Ramanujan saw, but we might somehow reach the same conclusion for a similar reason.

The only person I believe to have had as much talent for numbers is Gauss, who had the formal training such a mind needs from an early age. It seems reasonable to assume that Ramanujan might have changed the world of mathematics as vastly as Gauss did, had he been born under more propitious circumstances that afforded him an early start under expert guidance. These minds only require nudging in the right direction here and there. Euler began his higher education in mathematics at about thirteen when he went to study with the Bernoullis. One cannot say Gauss would have contributed more had he been given the opportunity to study with somone of Bernoulli caliber earlier on. The people instructing Gauss in his primary years were merely good not great, from all accounts. The duke of Brunswick had taken note of the young prodigy and made sure he was in a place of learning. There was a student or student-teacher six or seven years Gauss's senior interested in mathematics that the prodigy consorted with. This was perfectly enough. One can be quite assured that Gauss was equipped to solve any problem any mathematician up to his to his time and fifty years beyond ever solved, even those that were solved during his liftetime by others who were also great matheticians. What Abel and Galois did in proving there was no general method for solving equations of 5th degree and beyond, was a discovery for the ages, and it happened during Gauss's prime. Why didn't Gauss make that discovery?

One has only to look at how full the plate of Gauss was, at what he got done, to forgive him for leaving this major discovery out of his fireplace mantle collection. He was busy at that precise moment calculating in his head the orbit of Ceres from a few degrees of arc he had been given. It was an open problem in the world of mathematics, and one that had to be solved fast. Many were scurrying to calculate an orbit so the discovery of a new planetoid would not be lost. Luckily, there was already a computer in the world in 1822. When the time came only one set of calculations was correct. Ceres remergred from behind the sun where and when Gauss said it would. This was not the first time he had amazed the world, but this cemented his reputation for all time. While he was calculating the orbit in his head hre invented a new tool to assist him that we now call the Method of Least Squares. This is a tool now in universal use. Even Gauss could only work on so many things at a time. Few but Ripe, was his motto, remember. It is not true that he worked on only a few problems, but it is true that he brought most of them to ripeness that he did engage with. His mind was perhaps not superior to that of Euler but his method was cleaner and superior, I believe. Euler chopped down more individual trees than anyone. Gauss cleared forests and usually built a ranch on the spot. Euler did not build nearly as many ranches. That, I believe, is why Gauss is regarded a little higher, not only by myselgf but by mathematicians in general. None of this to detract from the legacy of Euler, who is one of my idols, but merely to point out that the talent of Gauss has probably not been seen in the world again except for perhaps Ramanujan, who had a pitiful start yet still made mighty contributions.

Euler was able to calculate to fifty decimal places in his head when he needed to. Gauss's solution was different--he simply memorized a book of logarithm tables in a day or two and the problem was taken care of. Now he could hold up for examination in his head any logarithms he wished to compare, and when he needed a logarithm it was right there at his disposal.

It is obvious I am too spent to discuss math right now, or I would be writing math instead of writing about mathematicans. It is good recreation. I will try again later to look at math.

[YesNo]

I have a hard time remembering five digits of pi. Memorizing a table of logarithms is probably out of the question for me. I have heard of people with synaesthesia who can see numbers as colors or shapes.

[desiresjab]

I have a hard time remembering five digits of pi. Memorizing a table of logarithms is probably out of the question for me. I have heard of people with synaesthesia who can see numbers as colors or shapes.

That guy from England, Daniel Tamet (not sure of spelling) has that synesthetic power, we might as well call it. Damage or deformation of the copus callosum which may have repaired itself the best way it could, seems to have something to do with the function of seeing numbers as colored shapes or smelling the notes of a flute concerto like a flower show.

I do not know if any synesthete ever turned his or her dysfunction into a mighty contribution in any of the arts or sciences. Interesting question. In which field would success be most likely? I suppose there are different degrees of the dysfunction, from light to full blown. Tesla might be candidate. Not sure. In the old days this ability is something that geniuses might have very intelligently hidden, for fear of being chained to a post while hags ran for kindling. Goethe had an unusual mind that grasped science differently than the standard model of the time. Could he have been one?

[YesNo]

They might have all been synaesthetic, but that may not be the correct word for it. Intuitive might be better. Or they were able to communicate better with their goddesses. Synaesthesia may be labelled a dysfunction but it appears to be just not normal functioning. I don't know if there is any brain correlates for it nor if there is anything wrong with it.

[desiresjab]

They might have all been synaesthetic, but that may not be the correct word for it. Intuitive might be better. Or they were able to communicate better with their goddesses. Synaesthesia may be labelled a dysfunction but it appears to be just not normal functioning. I don't know if there is any brain correlates for it nor if there is anything wrong with it.

That is why I italicized dysfunction.

[desiresjab]

Just for fun, and to do for factorials what is done for powers, consider the list below.

1!-0!=0
2!-1!=1
3!-2!=4
4!-3!=18
5!-4!=96
6!-5!=600

The recursive formula for a division of two consecutive factorials would be n!=(n+1)!/n+1. Very easy.

A formula for a subtaction of two consecutive factorials n!-(n-1)!, must look like this:

n!-n!/n, and after manipulation like

n!(n-1)/n= (n-1)(n-1)!=

n!-(n-1)!=(n-1)2(n-2)!. We also like the following from above because it is a factorial times a simple ratio, fast to work and succinct:

n!(n-1)/n.

* * * * *

To respond to my own question about why repeated iteration of the φ function on a number eventually reduces to a pure power of 2 before it has to by virtue of merely being even, after staring long at the Wiki-peja article on the function, one finds this statement, which not only proves that it does happen but shows why and when as well, if one contemplates:

φ(n) is even for n>2. Moreover, if n has r distinct odd prime factors, then 2r|φ(n). The vertical bar means “divides.”

* * * * *

Of the unsloved problems involving the φ function, Lehmer's conjecture looks captivating.

[YesNo]

To respond to my own question about why repeated iteration of the φ function on a number eventually reduces to a pure power of 2 before it has to by virtue of merely being even, after staring long at the Wiki-peja article on the function, one finds this statement, which not only proves that it does happen but shows why and when as well, if one contemplates:

φ(n) is even for n>2. Moreover, if n has r distinct odd prime factors, then 2r|φ(n). The vertical bar means “divides.”

* * * * *

Of the unsloved problems involving the φ function, Lehmer's conjecture looks captivating.

Lehmer's totient problem sounds interesting. It is the first I heard of it: https://en.wikipedia.org/wiki/Lehmer's_totient_problem I would be happy to understand how someone got as far as they did with that conjecture.

I think it makes sense that if there are r distinct odd composite primes dividing n then 2r divides the totient of n. More factors of 2 than r may divide it as well. I don't know what it means for a number to eventually reduce to a pure power of 2 before it has to by virtue of merely being even. One would have to find a measure for that concept, but one might exist.

[YesNo]

Regard Lehmer's conjecture, I can see why any such n so that φ(n) | n - 1 must be a Carmichael number. The order of any prime p dividing n would divide φ(n) and so divide n - 1 implying that pn-1[\sup] = 1 (mod n): http://mathworld.wolfram.com/LehmersTotientProblem.html

What about the other results?

(1) Why must it be square-free? (Edit: I can see why it must be square-free. If p[sup]r divides n then pr-1(p-1) divides n - 1. So p divides both n and n - 1 if r > 1. That implies p divides 1. So it must be square-free.)

(2) Why must it have at least 7 (or 11 or 14) distinct primes dividing it?

(3) Why must it be greater than 1023?

(4) Why must it be odd? (Edit: I can see why it must be odd. If n were even then n - 1 is odd. By assumption φ(n) | n - 1, but φ(n) is even for n > 2. Considering the two cases for n <= 2: For n = 2, n is prime and since the conjecture only applies to composite n, n = 2 does not count. Since 1 is not a composite number either, n = 1 is not covered by the conjecture either.)

Here's another reference: http://math.stackexchange.com/questions/173908/updates-on-lehmers-totient-problem

(5) Why are there no counterexamples of the form k2k + 1 (Here's the paper: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.770.6572&rep=rep1&type=pdf)

(6) Why are there no Fibonacci numbers as counterexamples?

(7) Is the following solution to the problem correct or not? https://www.youtube.com/watch?v=swbZqPjrcGk (Edit: This doesn't convince me. We can skip the powers of b since we know n must be square-free. So let composite n = bC where b is prime then if φ(n) = n - 1, we have φ(bC) = φ(b)φ(C) = n - 1 = bC - 1. So φ(C) = (bC - 1)/(b - 1). I don't see why that could not reduce to an integer. It would be an integer if C = 1 (more generally C = bk), but then n would be prime or not be square-free.)

[YesNo]

The more I am thinking of William Bouris' proof that Lehman numbers do not exist, the more I think there might be something to it. The basic idea is that φ(C) = (bC - 1)/(b - 1) cannot be an integer unless C = bk for some integer k which would imply that n is prime or n is not square-free. If that can be shown then there are no composite n = bC with b prime. That would be the same as saying bC is not congruent to 1 mod (b - 1). At any rate, this would be another way to look for a counterexample if one actually existed.

[desiresjab]

Lehmer's totient problem sounds interesting. It is the first I heard of it: https://en.wikipedia.org/wiki/Lehmer's_totient_problem I would be happy to understand how someone got as far as they did with that conjecture.

I think it makes sense that if there are r distinct odd composite primes dividing n then 2r divides the totient of n. More factors of 2 than r may divide it as well. I don't know what it means for a number to eventually reduce to a pure power of 2 before it has to by virtue of merely being even. One would have to find a measure for that concept, but one might exist.

Since I saw that all numbers did eventually reduce to a power of 2, the iterated function must necessarily reach a power of 2 merely by virtue of the function always being even. Why? There is no escape for the iterated φ function. We already know certain facts, do we not?

φ(φ(18))=2
φ(17)=16
φ(16)=8
φ(15)=8
φ(φ(13))=4
φ(12) above
φ(φ(11))=4
φ(10) above
φ(φ(9))=2
φ(8)=4
φ(φ(7))=2
φ(6) above
φ(5)=4, etc.

We already know these values and more for small values. A number descending from above in a function doomed always to spit back as output an even result, and whose last value is known to be 1, must eventually come to rest at 2, since that is the only argument for which the function will output 1. So it is quite forced for that reason, if for no other, and it turns out there is another reason, which returns pure powers of 2 as function values long before the hand of the function is forced to produce a power of 2 simply because its next value must be even.

[YesNo]

What I don't understand is what it means to "reduce to a pure factor of 2". The totient could only have one factor of 2 in it. Let n be a prime of the form 4m+3 to get a totient with only one factor of 2 in it.

[YesNo]

Regard Lehmer's conjecture, I can see why any such n so that φ(n) | n - 1 must be a Carmichael number. The order of any prime p dividing n would divide φ(n) and so divide n - 1 implying that pn-1[\sup] = 1 (mod n): http://mathworld.wolfram.com/LehmersTotientProblem.html

What about the other results?

(1) Why must it be square-free? (Edit: I can see why it must be square-free. If p[sup]r divides n then pr-1(p-1) divides n - 1. So p divides both n and n - 1 if r > 1. That implies p divides 1. So it must be square-free.)

(2) Why must it have at least 7 (or 11 or 14) distinct primes dividing it?

(3) Why must it be greater than 1023?

(4) Why must it be odd? (Edit: I can see why it must be odd. If n were even then n - 1 is odd. By assumption φ(n) | n - 1, but φ(n) is even for n > 2. Considering the two cases for n <= 2: For n = 2, n is prime and since the conjecture only applies to composite n, n = 2 does not count. Since 1 is not a composite number either, n = 1 is not covered by the conjecture either.)

Here's another reference: http://math.stackexchange.com/questions/173908/updates-on-lehmers-totient-problem

(5) Why are there no counterexamples of the form k2k + 1 (Here's the paper: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.770.6572&rep=rep1&type=pdf)

(6) Why are there no Fibonacci numbers as counterexamples?

(7) Is the following solution to the problem correct or not? https://www.youtube.com/watch?v=swbZqPjrcGk (Edit: This doesn't convince me. We can skip the powers of b since we know n must be square-free. So let composite n = bC where b is prime then if φ(n) = n - 1, we have φ(bC) = φ(b)φ(C) = n - 1 = bC - 1. So φ(C) = (bC - 1)/(b - 1). I don't see why that could not reduce to an integer. It would be an integer if C = 1 (more generally C = bk), but then n would be prime or not be square-free.)

I have skimmed through Lehmer's original paper, "On Euler's Totient Function": https://projecteuclid.org/download/pdf_1/euclid.bams/1183496203

There are many results in this paper. Bouris does not appear to have proved his result and the technique used of showing that a ratio cannot be an integer is used in Lehmer's paper. So I will skip (7).

[desiresjab]

What I don't understand is what it means to "reduce to a pure factor of 2". The totient could only have one factor of 2 in it. Let n be a prime of the form 4m+3 to get a totient with only one factor of 2 in it.

Let me do a better job of being clear. Sorry for the confusion.

Iterates to a pure power of 2.

[desiresjab]

I have skimmed through Lehmer's original paper, "On Euler's Totient Function": https://projecteuclid.org/download/pdf_1/euclid.bams/1183496203

There are many results in this paper. Bouris does not appear to have proved his result and the technique used of showing that a ratio cannot be an integer is used in Lehmer's paper. So I will skip (7).

I looked at the Bouris paper. One pass was really not enough, slow as I am. Almost every time he states that a proposition "assumes" something (at least fifteen times, it seemed) it would be necessary for me to think long and hard to verify his contention. I did not feel it was worth it, especially after glancing at the YouTube side menu where it seemed Bouris might have had other proofs of many famous propositions. He is obviously more than a crackpot, but I cannot go about verifying or unverifying every proof someone claims to have made of a famous proposition. The fact that he made this one in language I understand means I could follow it out, if I felt it was worth a prodigious effort. It was good mental exercise. To follow every detail completely would be too much exercise. A year from now I may look at a proof like this and follow it easily if past is precedent. I envy you if you can. But for now I will rely on your opinion of Bouris and marshall my strength for whatever takes me. I hear Liouville calling, yet I don't know. I also hear triangles calling, Euler calling, theories of categories and forms calling, class numbers calling...The great part about being retired and a math butterfly is that where I go is usually based on inspiration or the need to fulfill other inspirations.

I look at all links provided. Sometimes I have already read it. I often re-read things many times. If I find something I like, I will stay with it for days until I have drained it as well as I can.

One could spend forever looking at the connections of the Euler ф function--it is that centrally placed. One could find an involvement for it in practically any proposition. In short, what it does in number theory is stand in for the term p-1 in case of composite numbers. The rest of the time it is p-1. Armed with this idea and a few cogent connections, one may be able to go big game hunting in the wild and have a reasonable chance of spotting the beast in camouflage.

[YesNo]

I don't think Bouris has a solution. When I was searching for information on the problem his paper kept popping up so I had to consider it.

However, the Lehmer paper is worth reading. It contains the main ideas and proof techniques.

I found an old book on continued fractions by C. D. Olds that I started reading. This should help build a foundation for Liouville numbers.

[desiresjab]

I looked at the Lehmer paper too. I do not claim to have understood every ounce, though stylistically it was so much cleaner and easier than the Bouris to follow. For me it was a confirmation of my beliefs concerning the importance of style in mathematics. It is impossible to always be clear for those with less understanding of a topic, but clarity carries great weight as far as it can go, especially for readers. Lehmer's style takes this into account, that of Bouris did not seem to, at least for me it did not. A number theorist who became a master at explaining abstract concepts was H.L. Davenport.

[desiresjab]

I don't think Bouris has a solution. When I was searching for information on the problem his paper kept popping up so I had to consider it.

However, the Lehmer paper is worth reading. It contains the main ideas and proof techniques.

I found an old book on continued fractions by C. D. Olds that I started reading. This should help build a foundation for Liouville numbers.

Continued fractions are plumb scary!

[desiresjab]

In the meantime, though, I may be looking at decimal expansions, a topic clearly related to continured fractions.

[desiresjab]

I figured I would run into the DeuceHound formula, if I kept reading. See a close variant in the aritcle below used in a product. What is the formula for? The sum of the divisors of n, of course, usually denoted σ(n), proving I have not yet been around in the basic number theoretic functions as much as I need to be, or I would have recognized this. I will never forget it. The divisors functions (along with a few others), I have bascially ignored, but I have seen in the last few days that they are terribly well connected.

https://mathlesstraveled.com/2007/11/12/perfect-numbers-part-i/

[YesNo]

I liked how he put the latex math symbols on that wordpress page.

[desiresjab]

I liked how he put the latex math symbols on that wordpress page.

What?

[YesNo]

I didn't know Wordpress sites could format mathematics formulas the way that site formatted them.

[desiresjab]

I didn't know Wordpress sites could format mathematics formulas the way that site formatted them.

I have never even heard of WordPress.

[Dreamwoven]

Wordpress have their own blogs, I have wordpress blogs and they allow public re-blogging of posts, which is very handy. See this: https://wordpress.org/news/

[desiresjab]

Wordpress have their own blogs, I have wordpress blogs and they allow public re-blogging of posts, which is very handy. See this: https://wordpress.org/news/

How much do they want for it?

[YesNo]

You can set up a Wordpress blog for free. You can also get jupyter notebooks for free which lets you format using mathjax which I think is the same code. When I posted links to the jupyter notebooks before I was using that same code to generate those math symbols.

[desiresjab]

You can set up a Wordpress blog for free. You can also get jupyter notebooks for free which lets you format using mathjax which I think is the same code. When I posted links to the jupyter notebooks before I was using that same code to generate those math symbols.

I can do almost everything with my OpenOffice word processor. The thing I cannot do is get subscripts and supercscripts to line up correctly when I want to use both on a sigma, for instance.

[YesNo]

What I use are Google docs and sheets for my personal documents. You just need a Google account to get that. It is all in the browser. You can also use mathjax with it after installing a plugin, but I use jupyter notebooks for mathematics with the underlying python kernel so I can calculate right in the notebook. I also use Google to back up all my photos on my phone as well as copy them to my computer (Windows 10).

I was reading more about Lehmer numbers. Any Lehmer number is also a Carmichael number. I can see why Carmichael numbers exist and Lehmer numbers probably don't. The Carmichael function lambda(n) is smaller than the totient, phi(n), and so it has a better chance of dividing n - 1. For example, the Carmichael number, 561 = 3*11*17, each of 2, 10 and 16 divide 560, but not their product.

[desiresjab]

Right now my head is fpinning in amazement at the fimple divifor functions, which fhow up in all kinds of not-fo-fimple places. They are involved in fome bigtime formulas by powerhoufe mathematicians and even have a clofe connection to the Reimann hypothefis.

We worked out σ1(pk) for ourfelves, and may have gotten to the more general formula if we had kept at it. The fimple functions are pure magic, but one fhould not be amazed at them, for they are there to be understood and are among the more underftandable objects in number theory. The mulitplicity of their connections ftill dazzle. But one can ftare at each one of them and fully underftand why there is a function there. We ftared fo long at σ1 that we know exactly how it works, we have taken the myftery out of it. I have realized I need to ftare now at what Wiki-peja calls σ0. I have not been working or thinking much becaufe I am coming out of a depreffion. Oh, by the way, of courfe one fees many obvious connections of thefe functions to the Euler phi function, which the article explores. The formulas are fuddenly no longer fimple, they look like ftuff Ramanujan himfelf would have worked on or produced in this field, and indeed he and Hardy were working in the immediate area. I will feel much ftronger once I tie up the divifior functions. I am impatient but ftill recovering, for I want to be off to the theory of lower bounds in logarithms. The more myftery I take out of thefe things the better I might feel about it when I have to die.

[desiresjab]

I have forgotten fome of the techniques for calculating limits. I fuppofe reviewing them, then, had ought to be a profitable venture before I look at the theory of lower bounds in logs. I remember there were fome functions you could not tell fimply by looking at whether they converged or not until fome proper manipulation had been done. It fhould take an hour to review what is proper. But when will I get to that hour, being as lazy as I am ambitious yet full of fchemes for learning? I tend to circle thefe propofitions flowly like a dog fizing up its rival, once I have them in my fights. Then I rufh in fnarling.

[desiresjab]

The proof in this link makes it cryftal clear why there can be a function for the number of divifors of a number. The inconvenience is we ftill have to break it down to prime factorization, the number's pretty face is not enough to give us the number of divifors it has. I knew thefe functions were fimple. Why did I ftay away fo long? Juft lazy or afraid. Cryftal clear. Aren't there only two important ones--the number of divifors and the fum of the divifors? It feems like there was another. Maybe I juft got my notations croffed up.

http://mathschallenge.net/library/number/number_of_divisors

[Danik 2016]

Lol! Some "f" spam here.

[YesNo]

It does seem that the "s" becomes an "f" often, but I have had things like that happen with a faulty keyboard. Sometimes it is just my fingers.

I agree that multiplicative functions are convenient until factoring the number becomes difficult. In the case of that divisor count function one would need a full factorization to take advantage of it.

[Dreamwoven]

It gives a very nice lisp-like feel to the text!

[Danik 2016]

It does seem that the "s" becomes an "f" often, but I have had things like that happen with a faulty keyboard. Sometimes it is just my fingers.

I agree that multiplicative functions are convenient until factoring the number becomes difficult. In the case of that divisor count function one would need a full factorization to take advantage of it.
I thought it might be a joke and reacted accordingly. But I think you are right. It must be a keyboard problem.

[desiresjab]

I thought it might be a joke and reacted accordingly. But I think you are right. It must be a keyboard problem.

No, all I am doing is mimicking 17th century writers. Newton and some of the people he communicated with used f's in the places of s's sometimes, but only with rules. I never figured out why they did this. Maybe someone on here knows. I think the rule was at the beginning and in the middle of words but not at the end. It may just be something idosyncratic that developed. It was probably just an elongated s. I don't know the reason. No, folks, I have not flipped, unless you think I already was.

At least I found out who reads the thread. I already thought Danik and DW were.

[desiresjab]

Now for the simplest connection of all between ф and σ.

ф(n)=σ0(n) where n is a prime. Makes perfect sense, right? The number of divisors, including itself, is equal to the number of numbers less than or equal to it which are not relatively prime to it.

The number of divisors of a prime is just two, itself and 1. Two is the number of elements less than or equal to n which are not relatively prime to it. The two functions are, then, identical in the case of a prime.

I just wanted to throw that in for the viewers. I think it is important and easy to remember, and knowing it might allow you to derive other related functions if there was a need in the wild. Number of divisors and number of numbers not relatively prime to it, are the same thing for a prime. That is where ф will always equal σ0.

By the way σ0 is the divisors of a number. Any divisor is worth 1 in the count of divisors. Whereas σ1 designates the sum of those same divisors.

[desiresjab]

The reason I am dwelling on the σ functions is I now believe they are the most critical functions of traditional number theory. Even the ф function is merely a type of shorthand for σ, it now seems to me. The ф function, which itself is extremely important is really just shorthand for manipulations of the divisor function. That makes the divisor functions incredibly important, when ф is just a special case of them!

Is this vision correct? I believe so, but perhaps I am missing something.

[desiresjab]

Mathematical formulas in books are indeed succinct and exact, but for that reason all the more difficult to see in their stark simplicty sometimes. Some mathematical propositions that could require great work to unravel without particular insights into this simplicity, are intuitively taken in at a glance with these insights under one's belt.

Let us examine the law that says if two numbers do indeed share a common factor, then that factor will also be a factor of their difference. If A and B share a factor that factor will also divide A-B.

Let us call n the common factor between A and B. Then pn=A and qn=B. But why the general language? Let's get specific. That is what we came to do.

A=6n, B=4n. Now isn't that better? Now there is no problem. Yes, we see easily that the common factor n still divides 6n-4n=2n, the difference of A and B.

When the general language is removed from mathematical propositions, it often serves intuitive clarity. This is the goal. This clarity is exactly what we want. I believe it cannot be possessed at the higher levels until all the elementary propositions are solid as rock in the mathematician's mind. This is why we linger, and not without profit.

[desiresjab]

Philosophical mood today. Thinking about the history of mathematics. If east injuns 1500 years ago were smart enough to devise and consider what we mistakenly today call Pell's equation (x2-ny2=1), then it would be foolish to believe they did not have possession of propositions in number theory which are more elementary. They did not leap to a proposition like Pell's equation from pure ignorance any more than we would be able to do so today without massive preparation and a solid foundation to build upon. In other words, they had possession of most of what we call elementary number theory.

Perhaps the ancient Injuns and Chinese did not yet have possession of abstract algebra and group theory; perhaps they had no theory of quadratic forms. Then again, we really do not know what some isolated individual might have acheived in his hut, those results being lost forever with his death. By the time European mathematcis began its heyday, Europeans had extreme advantages in communication that had been available to no others before them. Discovered knowledge was cabable of dissemination as never before. What one man discovered, many others had the chance to review and build upon, knowledge no longer had to die in a cave with its discoverer.

We only know of some ancient societies that had this mathematics. We know for certain that many did not. In a pure white landscape it is easy to imagine why Eskimos would not develop even a counting system. North American injuns were mostly not advanced at all, but some of the southern injuns like Mayans and Inca were well advanced, and obviously had possession of some mathematics. This means to me that there were probably some smart Inca holed up somewhere doing number theory for curiosity and pleasure. Sub Saharan Africa seems to have had no advanced cultures. Along the Nile, Tigris and Euphrates there were numerous advanced civilizations. I believe we can assume that one mark of advanced societies is at least the mathematics of measurement and the beginnings of number theory.

[desiresjab]

My loquacity today seems to have no upward bound. I must be recovered..er.. I mean I am on the other side of the cycle now..ahem! Recovered, I said.

A thought keeps nagging me. Despite the difficulty of number theory, we do not know much. By we I mean mankind. It looks like we know a lot, but I am beginning to see it in such a way which means we do not actually know much. Now, we do know a lot of detail that depends on a few propositions that may be intertwined silently in our formulas making them valid wherever we take them. That these silent and critical propositions are so few in number at the base of our structure is what I mean by us knowing not so much. Yes, there is quite a bit all right, but not as much as it will at first seem. Wherever we could relate these critical propositions we (mankind) have delved deep on the spot. The difficulty of number theory lies less in its breadth than its depth. Group theory, for instance, comfortably encompasses number theory (as well as other forms of mathematics), not the other way around.

Once one knows these critical drawstrings (eyebrows hunched conspiratorily) one can see how they are tightening up numerous other propositions. I may be oversimplifying, but make no mistake number theory is difficult and is known to be, it has that reputation given by the masters themselves.

What I am gaining is probably an overview. Everything goes back to a few propositions, and if they were not true then none of the further explorations would be either. Deep propositions always keep a tether line to simpler ones, is another way, perhaps, of expressing the same thing. Once you know what they are tethered to, the understanding of overview begins to set in, must be what is happening in my brain.

* * * * *

Allow me to correct a small notational error from earlier. I said that for prime numbers that the anti ф(n) and σ0(n) were the same, but ф(n) is defined as the numbers less than n which are relatively prime to it, not less than or equal to it.

I have never heard of such a thing, but we could define this anti ф(n) to equal n-ф(n).

Then [anti ф(n)]+1=σ0(n), where n is a prime number, and the connection remains between ф and σ.

[YesNo]

The totient or ф(n) where n is prime would be n - 1, that is the number of integers relatively prime to n and less than or equal to n. If n is prime all n - 1 of them are relatively prime to n. This article also defines a cototient as n - ф(n). That should be 1 for n prime since only n would have a prime factor, itself, in common with n. https://en.wikipedia.org/wiki/Euler%27s_totient_function

The σ0(n) where n is prime would be 2, namely, 1 and n. Those are the only divisors of n if n is prime. https://en.wikipedia.org/wiki/Divisor_function

Edit: I just saw your most recent post which I think is saying the same thing.

[desiresjab]

The totient or ф(n) where n is prime would be n - 1, that is the number of integers relatively prime to n and less than or equal to n. If n is prime all n - 1 of them are relatively prime to n. This article also defines a cototient as n - ф(n). That should be 1 for n prime since only n would have a prime factor, itself, in common with n. https://en.wikipedia.org/wiki/Euler%27s_totient_function

The σ0(n) where n is prime would be 2, namely, 1 and n. Those are the only divisors of n if n is prime. https://en.wikipedia.org/wiki/Divisor_function

Edit: I just saw your most recent post which I think is saying the same thing.

They call that a cototient, eh?

[desiresjab]

Pell's equation is an object worth pondering.

x2-ny2=1,

is the equation of a hyperbola. Remember those conic sections you studied somewhere along the line, folks. This is an equation for one of them. It is not really a Diophantine equation. But we simply make it so by declaring we are interested only in its integer solutions. We already know n is an integer, we insist that x and y be integers as well, that is all. When we look at the ratio y/x, we see that successive solutions give better and better approximations of √n. It is one way to get the square root of a non square integer.

If we wanted the square root of 2, we would substitute 2 in the equation for n:

x2-2y2=1. The ratio of x and y in the integer solutions of this equation will now give better and better approximations to the square root of 2, as x and y grow larger.

Besides being a conic section, which we are know are important continuous functions, this equation leads a double life as a Diophantine, where it can do one little job a continued fraction does in a fraction of the time, I assume.

[YesNo]

Here's more on Pell's equation: https://en.wikipedia.org/wiki/Pell's_equation

I found an old book by Ivan Niven, "Numbers: Rational and Irrational", that has two final chapters on rational approximation of an irrational number and a proof why the Liouville number is transcendental. I am going through the exercises on those two sections to make sure I understand it. The earlier chapters are elementary. I think it was written for high school students, but even elementary material can be illuminating.

[desiresjab]

Here's more on Pell's equation: https://en.wikipedia.org/wiki/Pell's_equation

I found an old book by Ivan Niven, "Numbers: Rational and Irrational", that has two final chapters on rational approximation of an irrational number and a proof why the Liouville number is transcendental. I am going through the exercises on those two sections to make sure I understand it. The earlier chapters are elementary. I think it was written for high school students, but even elementary material can be illuminating.

Sometimes elementary material can be very illuminating. Maybe this is because a lot of the formality has been stripped away, making the essential ideas easier to see.

[desiresjab]

What I use are Google docs and sheets for my personal documents. You just need a Google account to get that. It is all in the browser. You can also use mathjax with it after installing a plugin, but I use jupyter notebooks for mathematics with the underlying python kernel so I can calculate right in the notebook. I also use Google to back up all my photos on my phone as well as copy them to my computer (Windows 10).

I was reading more about Lehmer numbers. Any Lehmer number is also a Carmichael number. I can see why Carmichael numbers exist and Lehmer numbers probably don't. The Carmichael function lambda(n) is smaller than the totient, phi(n), and so it has a better chance of dividing n - 1. For example, the Carmichael number, 561 = 3*11*17, each of 2, 10 and 16 divide 560, but not their product.

Carmichael numbers are such unweildy beasts it would be semi-amazing that anyone found an example before the age of computers if not for individual examples of human industry and tenacity that far exceed the effort needed to find one of these. Euler and Gauss did incredible calculations in their heads. But still, Carmichael numbers were not predicted by theory, were they? What were those industrious early explorers who found examples looking for? I mean, why were they looking? What was there for them to have faith in, since no theory I am aware of said there would be composites which behaved just like primes when put through the machine of Fermat's Little Theorem? It quite intrigues me. I suppose they were looking because no theory said there could not be such composites. Or, if at that time Fermat's Little Theorem was believed to possibly be a reliable test for primes, it makes sense that some eager beavers would have been engaged in the pursuit of a counter example. What about that?

[YesNo]

I wonder if there are shortcuts to determining if a number is a Carmichael number?

I don't know the history, but there are many composite numbers that are pseudoprimes to one base or the other. All odd composites are pseudoprimes to base 1 and n - 1 since 1n-1=1 mod n and (n-1)n-1 = -1n-1 = 1 mod n where n is odd. It seems to make sense to look for something like this: Can one find a composite number that is a pseudoprime to every base relatively prime to the composite number?

Carmichael had Korselt's criterion (1899) to lead the way before he found one in 1910: https://en.wikipedia.org/wiki/Carmichael_number

[desiresjab]

I wonder if there are shortcuts to determining if a number is a Carmichael number?

I don't know the history, but there are many composite numbers that are pseudoprimes to one base or the other. All odd composites are pseudoprimes to base 1 and n - 1 since 1n-1=1 mod n and (n-1)n-1 = -1n-1 = 1 mod n where n is odd. It seems to make sense to look for something like this: Can one find a composite number that is a pseudoprime to every base relatively prime to the composite number?

Carmichael had Korselt's criterion (1899) to lead the way before he found one in 1910: https://en.wikipedia.org/wiki/Carmichael_number

Merry Christmas to you, good sir, and to everyone reading.

I see why from the beginning the search was on for the beast eventually named Carmichael number. Many (all?) composites have a subset of their residue system consisting of "some" of the residue classes (more on "some" later). This special subset of numbers ({q1, q2, ...qk}) when raised to the n-1 power leave a residue of 1, just as numbers would do under a prime modulus. These q's are called false witnesses or liars.

No composite number n was known for which all its n-1 residue classes were false witnesses. That is precisely what a Carmichael number is.

Raising every number between 1 and 561 to the 560th power was quite a bit of work in the days before computers, to find that 561 was a Carmichael number. Smart investigators, however, would have known, I suspect, that after surpassing a "certain number of" (more on "certain number of" later) false witnesses in their calculations, 561 was a Carmichael number, precluding the necessity of doing all 559 calculatiions which are not trivial.

[YesNo]

It is not all n-1 residue classes that are false witnesses to make a Carmichael number. Only those that are relatively prime to n. In the case of a Carmichael number, which are squarefree, one would have to get a factor for Fermat's criterion to be accurate, but it wouldn't be accurate for an = a (mod n). That would still work.

Consider 561 = 3*11*17, a Carmichael number (assuming the python is correctly programmed):

3561 = 3 mod 561, but 3560 = 375 mod 561
11561 = 11 mod 561, but 11560 = 154 mod 361
17561 = 17 mod 561, but 17560 = 34 mod 361

There are seem to be at least three layers of tests each restricting the exponent of the witness a bit more:

an = a mod n
an-1 = 1 mod n Fermat's test
a(n-1)/2 = (a/n) mod n Euler's test

[desiresjab]

It is not all n-1 residue classes that are false witnesses to make a Carmichael number. Only those that are relatively prime to n.

Yes, of course. You have to excuse me-- I am so used to working with prime moduli that I sometimes unconsciously lapse into that mode. For a Carmichael number all numbers in ф(n) must bear false witness.



In the case of a Carmichael number, which are squarefree, one would have to get a factor for Fermat's criterion to be accurate, but it wouldn't be accurate for an = a (mod n). That would still work.

Consider 561 = 3*11*17, a Carmichael number (assuming the python is correctly programmed):

3561 = 3 mod 561, but 3560 = 375 mod 561
11561 = 11 mod 561, but 11560 = 154 mod 361
17561 = 17 mod 561, but 17560 = 34 mod 361

There are seem to be at least three layers of tests each restricting the exponent of the witness a bit more:

an = a mod n
an-1 = 1 mod n Fermat's test
a(n-1)/2 = (a/n) mod n Euler's test

I think I see all that. I would not have been able to do your three lines of calculations to get those results without a calculator. I suspect I should be able to do it through manipulation alone. How did you get those values?

Okay. I think I see much of what is going on with Carmichael numbers. The thing I do not see is why a Carmichael number was eventually inevitable. If it happened on the number line, it was inevitable. Do you see why it was inevitable? What would one look at in Euler or Gauss's position to know a Carmichael was inevitable if you went out the number line far enough?

[YesNo]

There is a lot I don't intuitively see about Carmichael numbers. I don't know why they should exist, but there is a proof that infinitely many of them exist. I haven't read that proof and I don't even know how I would try to show something like that.

As far as getting those values, I put them into a jupyter notebook, created a function and ran the function. I wouldn't trust the result without checking it there. I don't have a proof that all Carmichael numbers should behave the same way, only that 561 does.

def car(base,num):
return([(base**num)%num, (base**(num-1))%num])

[desiresjab]

There is a lot I don't intuitively see about Carmichael numbers. I don't know why they should exist, but there is a proof that infinitely many of them exist. I haven't read that proof and I don't even know how I would try to show something like that.

As far as getting those values, I put them into a jupyter notebook, created a function and ran the function. I wouldn't trust the result without checking it there. I don't have a proof that all Carmichael numbers should behave the same way, only that 561 does.

def car(base,num):
return([(base**num)%num, (base**(num-1))%num])

I would like to ask some questions about your academic career. It sounds like you were a math major, since you have degrees there. What was your emphasis? I assume you are at least familiar with real analysis, complex analysis, differential equations (though you may have forgotten some due to not using), whereas all I ever had beyond college algebra, trig and analytic geometry was one year of calculus. There are times I definitely feel the desire for more formal education in math.

[YesNo]

Yes, I remember taking classes in all of those subjects. My main interest was computational problems and hence the focus on data analysis. I can remember not liking the big-O notation which estimated an upper bound on solutions as n got large. I wanted to know how many solutions there actually were and how fast could one compute them.

The formal education has advantages. It opens up job opportunities. It gets one in the habit of writing in a certain way. It focuses attention on research papers.

However, just having someone to talk to about these topics is very useful. I would not even be thinking about them now, if you weren't bringing them up. That is where someone who is an academic would have an advantage over both of us. Not only are they trained but they are among a community who are trying to publish new research.

[desiresjab]

I want to look at the repetends of decimals. Ay first glance it seems there would be a complete theory, but I think not.

We can say something about decimal numbers before they are even calculated, however. We are able to say certain things. If a prime number which ends with the digit 7 has a period of p-1 when decimalized (1/p), the digits in its decimal will have an equal number of the digits from 0 to 9 except that the six digits in the expansion of 1/7 will occur one extra time.

I suspect something similar can be said about primes ending in other digits, but have no proof yet or even a demonstration.

The period of the decimal expansion does not appear to coincide with anything familiar except the factors of n. Which factor on sight the period will emulate seems to be a problem of depth.

For 37, the period is only three, which is neither the least or the greatest factor of 36. Yet 31 has a period of (p-1)/2, which is the greatest factor of 32 smaller than 32. This is not some strange number to us.

Like everything else in number theory, I suspect knowledge of repetends is not complete and ceases somewhere--to be precise, right where our inability to completely master primes begins. This will be a very nice surprise if you can inform me otherwise here.

P.S. It does not agree with the order either, the order of 4 being three.

[YesNo]

Here's something on decimal expansions. I have only read the first page: http://people.csail.mit.edu/kuat/courses/decimal-expansions.pdf

Continued fractions also have repetitions that might be useful.

[desiresjab]

Here's something on decimal expansions. I have only read the first page: http://people.csail.mit.edu/kuat/courses/decimal-expansions.pdf

Continued fractions also have repetitions that might be useful.

I found that a pretty tough paper. One expects no less at the graduate level. On the other hand I see most of what is going on in it. Primitive roots mod (p) appear to have periods p-1, if I interpreted it correctly. Of course we already knew that about primitive roots, at least what their period would be, powered up. The good part is that behavior carries over to the decimal expansion.

[YesNo]

I haven't had time to finish it. I am wondering how they will prove the main result on the first page that given an integer d and a base a (not equal to 2) there is a prime p such that the length of the period of the expansion of 1/p is d. I would not think this would work for any d and it doesn't seem to work for base 2. Supposedly there is no prime p that has period of length 6 in base 2.

[desiresjab]

I guess it is a curious fact:

(6k+1)(12k+1)(18k+1) may be a Carmichael number whenever all three factors are prime, is how I took it. Is it sufficient but not necessary? The curious fact is that 561 is not of this form, for it is (3)(11)(17). There must be more than one breed of Carmichael number. It goes to show that in the deep structure of numbers things are never quite simple. I may never know why a Carmichael number just had to be, or whether in the time of Euler and Gauss it was highly suspected there were such beasts. I do not see 561 defeating either of that pair's calculating ability, so perhaps they were not too suspicious in their day. Was Fermat's little theorem considered a good primality test then? I do not see either of them falling for that, especially Gauss, who only showed winning hands. It is kind of a nice historical question apart from understanding the technically difficult parts.

As I look at the complex formulas for Carmichael numbers and repetends, I see familiar friends like the G.C.D and the phi function filling them out. Sometimes there is that big O term you were talking about, the error term, which comes last and expresses the probablility of error. I think Hardy and Littlewood made a formula for the density of Carmichael numbers, but it could be another formula I am confusing it with.

The link you provided for the repetends study was pretty deep. I did not like their style. I think by going over it quite a few times I can extract most of it. I might find some other articles dealing with the same subject.

I am going to need some work on why i and e and pi are in those formulas, in fact I have noticed they seem to be a staple in many of the higher number theoretic formulas I am seeing. I have to get comfortable with that. Gauss's criterion for recognizing a mathematician of the first class in the making was immediate understanding of Euler's formula eiπ+1=0. I do not qualify. I have to find out why these terms have found a home in the number theoretic formulas. I believe it must be through the complex number system, and I know some trig identities are involved.

[desiresjab]

Oh boy! I think I found the key statement I was looking for. It was in a Wiki-peja article.

The length of the repetend of 1/p is equal to the order of 10 (mod p). If 10 is a primitive root mod p, the reptend length is equal to p-1; if not, the repetend length is a factor of p-1.

Just how precisely those fractions which produce lengths which are factors of p-1 can be nailed down, I am not sure. It may be something found in the link you provided for repetends. But that factor which is the length of the repetend is merely the order of 10 (mod p), right? That says it all. Those are the bare facts, the rest is just proofs.

[YesNo]

That sounds like a good distinction. I don't understand why the base needs to be a primitive root mod p, but it should have something to do with it.

I think there are Carmichael numbers with more than three prime factors, but not less than three.

One doesn't have to understand everything in an article, nor even read all of it. I rarely finish reading things.

The i, e and pi are in the cyclotomic polynomials to get roots of xn - 1 in that article. I think that is just one way to get uniformly spaced points around the unit circle. One could use sines and cosines, but this is more compact. https://en.wikipedia.org/wiki/Euler's_identity I don't have an intuitive feel for this either. It is just a way to calculate. It is kind of like quantum physics. One can calculate and get useful results without knowing what it is one is talking about.

[desiresjab]

That sounds like a good distinction. I don't understand why the base needs to be a primitive root mod p, but it should have something to do with it.


I believe because anything but a primitive root will power up to 1 (mod p) before its p-1 power. The power at which it reaches 1 for the first time is the order. The order of 10 (mod p) is the length of the repetend. Of course 10p-1 is not exactly easy to calculate for large primes. After one had gone as far as (p-1)/2, the largest possible factor of p-1 except for itself, one could safely conclude they were dealing with a primitive root (mod p).

[YesNo]

Yes, I see that now. The primitive root will be able to generate all relatively prime values less than the prime p and there are p-1 of them.

[desiresjab]

What I do not see is the structural similarity between various types of Carmichaels. That (6k+1)(12k+1)(18k+1) criterion was of course for a three-factor Carmichael. But that three-factor Carmichael (3)(11)(17) is not of the above form, and I have no idea whether an infinite number of Carmichael numbers are of a different form than the one above. So what form is it? I have not located the unifying principal between all Carmichael numbers. There are infinitely many of them, and even infinitely many Carmichaels with any number of factors you care to name. It seems to me there has to be some principle unifying all Carmichael numbers. It is probably sitting right in front of my nose and I cannot see it. I will see it, at least I think so presently.

[desiresjab]

This unifying principle between all Carmichael numbers of any form, may be what the Chinese civil servant recently tapped into. I feel it is tappable. I feel you and I have to tap it now, since we went ahead and challenged it, asking our not so innocent questions, and we are sure to be named wusses if we back away now without the answer. We did not back away from quadratic reciprocity until we knew there was at that time no more getting to be had from it with the tools we were using. I dread these prolonged struggles because I am lazy and always look for an easy way, in slovenly accordance with Occam's razor.

It would certainly be nice if 561 were a single rogue example and all other Carmichaels were of the above form. I highly doubt that, but do not yet know that it is untrue. I have a heuristic theory of rogue solutions underway right now which I hope to present ri'cheer in the near future.

[desiresjab]

A more careful reading of Carmichaels reveals that (6k+1)(12k+1)(18k+1) produces a subset of Carmichaels when all three factors are prime. This was proved in 1939. It is not yet proven that this is an infinitely repeatable subset, but highly suspected.

The fact that there are infinitely many Carmichael ideals sounds a bit prohibitive for finding one nature that defines them all. I do not know enough about the theory of ideals yet. There may not be an approach to them that is not laden with abstract algebraic notations. Perhaps we should proceed as if there were one nature to be found, until otherwise is shown which we can recognize.

[desiresjab]

And here is a highly provocative statement from Wiki-peja:

Since infinitely many prime numbers split completely in any number field, there are infinitely many Carmichael ideals in OK. For example, if p is any prime number that is 1 mod 4, the ideal (p) in the Gaussian integers Z[i] is a Carmichael ideal.

[YesNo]

Which Wikipedia article were you reading?

[desiresjab]

Which Wikipedia article were you reading?

I l,ooked through the stuff I read. Could not find the quotes I gave. It was one of the following links. The one on category theory is killer abstract. It is hard to find a good entry point for any one of these subjects because they are all inter-related and used in the definitions of each other.

https://en.wikipedia.org/wiki/Category_theory

https://en.wikipedia.org/wiki/Injective_function

https://en.wikipedia.org/wiki/Ideal_class_group

https://en.wikipedia.org/wiki/Carmichael_number

https://en.wikipedia.org/wiki/Ideal_(ring_theory)

https://en.wikipedia.org/wiki/Abstract_algebra

https://en.wikipedia.org/wiki/Quotient_ring

https://en.wikipedia.org/wiki/Fundamental_theorem_on_homomorphisms

https://en.wikipedia.org/wiki/Surjective_function

[YesNo]

Those articles like a good starting point. I haven't read read all, but I will start with the one on Carmichael numbers.

[desiresjab]

Those articles like a good starting point. I haven't read read all, but I will start with the one on Carmichael numbers.

I will be going over them again and again, lad, combing for details I can uptake. The only thing I can see which unifies all Carmichael numbers is the criterion that the p-1 of all primes dividing n also divides n-1. That seems to be it. It would be wonderful to find more connections unifying them. There might be some other unifying principle which would significantly lessen the work involved in computer searches for them.

* * * * *

Several more things to mention. The first is the difficulty of the math now facing us. How difficult for you I do not know, but for me quite difficult are the great abstract fields ahead. There can hardly be enough preparation. I am like a wolf that has been picking off theorems from the outskirts of the herd for years. I rather know what to expect and still find the slogging torturous. Those articles created far more questions than they answered for me. The basic idea of an ideal seems simple enough at first presentation. The idea of a field consisting of multiples of single number such as 2 or 3, is easy to grasp. For instance the even numbers. Every multiplication or addition of any two elements in the field produces another element of the field, a basic requirement of groups. Easy, right?

But then one learns that Carmichael numbers also form an ideal field (my language is not yet straight). And one says, What? I thought ideals were simple multiples of a single number. Carmichael numbers are not multiples of one another that I can see.

So I have a long ways to go to get the basics of this higher mathematics under control. If Category Theory is not higher mathematics, then I do not know what is. If Category Theory is still elementary mathematics, I am, sir, a moslem's uncle. From what I have seen so far it offers the highest and most abstract view available in mathematics. I have been toiling in the grease of the gears of raw numbers for too long. I now must attempt to take the step to the lofty views which dash away whole categories with a few slashes of chalk.

[desiresjab]

The other topic today is the prevelance of mathematical genius. I hope some who normally only read here will pitch in with some ideas.

Are there talents equivalent to Gauss, Euler, Newton and Ramanujan in the world today? If not, why not?

There are more people in the world than ever and that means more researchers than ever. Statistically, we should have math men as great as those aforementioned in the world today. But one gets the feeling a Gauss or Newton might have already completely realized the mathematics of string theory, tying it to physics and forcing the paradigm shift in thinking. There are great mathematical minds in every age. Of course we have them now, too.

Fifteen and seventeen year old thinkers do not make major discoveries anymore. They might make a small one here and there. Perhaps the period of preparation needed to bring oneself up to specs on contemporary research topics is so long it precludes that happening anymore. At eighteen one could make a reasonable argument that Gauss and Newton were already the world's best mathematicians. I doubt if this is still possible, but I hope it still is. I would love to see some fifteen year old force the world to shift its paradigm of reality.

[desiresjab]

One limiting factor of mathematics is the lack of poetry. Yes, I said lack of poetry. For poetry consists always of employing one word in a context where it is superior to all others. Mathematicians are decent at choosing appropriate symbols, but are often worse than mediocre when it comes to choosing terms in a spoken language, and downright awful for accepting as standard some of the terms they have.

Either the term order refers to the power (mod p) to which one must raise the constant a until the value wraps back around to one, or it refers to the number of elements in a Finite Galois Field. Which is it, math boys? You boys do not even care that these two branches are closely aligned, and expect newcomers to put up with your perfectly avoidable ambiguities as if they did not exist. Ignore them, you teach.

Let the student instruct the master, the master whose purpose is to eliminate ambiguity. In how many ways in how many branches must you idiots continue to use the words Order, Class, Congruent, et al, to mean different things? I can find such examples all over mathematics.

* * * * *

While we are at it, I do not think the symbol for pi had ought to be used for anything in math other than the ratio of a circumference to a diameter. Admittedly, there is a genuine paucity of available symbols not found in the actual language where these symbols combine to form words, many of which are synonyms, so there is more excuse for bad notation in math than bad poetry where a multiplicity words are available.

The notion will make more sense to me (in Group Theory) if it turns out that it equates to the power we must raise a to (mod p) before it becomes a again. This would be just another angle on Fermat's Little Theorem, the unreduced version where ap≡a (mod p). Is that what is going on with the confusion of words mathematicians have chosen? I could accept that linguistic disparity because both notions of order are at least based on the same theorem in two different forms, those two forms being:

ap≡a (mod p), and ap-1≡1 (mod p).

[YesNo]

I don't know much about category theory.

We have been traveling to avoid the cold in Chicago, although it was rather warm there last week. I just got the computer connected to the internet again.

There are a lot of overused words in mathematics as you mentioned like "order". Whatever they refer to in some context should be unambiguous or one won't be able to use logic to show something is false or true.

[desiresjab]

And speaking of ugly and unimaginative...the slash /, should never be used in mathematics to mean anything other than simple division. In the group theory I now need to look at they seem to use it all the time for something else. I hate this. I detest it. It makes mathematicians look stupid in their own way.

[desiresjab]

While we are waiting for formal higher math to reveal the best port of entry for my weaknesses, let us take a look at the Euler phi function, with the eventual purpose of understanding precisely a modrern encrytion system such as RSA through this function. We looked at this function before but it always deserves more. I do not remember what we said before.

The function is always even. For an even number it is easy to see why by inspection of this formula for ф(n):

ф(n)=nΠp|n(1-1/p).

Notice the right side is multiplied by n, which we already know is an even number. Case closed.

When n is odd, that is where there are no powers of 2 in its prime factorization, the following formula for a power of a prime makes the answer transparently obvious:

ф(pk)=pk(1-1/p).

That subtraction on the right must result in an odd number minus an odd number, which we know is always even. Case closed. The ф function is always even.

* * * * *

I am having a difficult time penetrating very far into ideal numbers. I can see it will take more than a few new tools. One thing about the ф function is that it is always even. I may be a dummy, but I believe that allows me to say the function could be stated in terms of the ideal of the even numbers. The function does play a significant role in ideal theory, if I am not mistaken. No surprise there. That is what great functions do--show up everywhere.

The elementary functions are so interrelated one could probably state any of them in terms of the others. The Euler ф function would be no more than a special and quite extended exercise or circumstance of the divisor function. That would be quite a mess. Thankfully, there is a division of labor among a bevy, a cluster, of closely related elementary functions, parcelling out the various applications and implications. This is much nicer.

I can already tell anyone riding along and reading who wants to follow the mechanical details of the encrytion process, that it will boil down to some tricks of exponents on ф, almost magic-like tricks, where you have to follow the bouncing ball. Even after you see it, it can get away from you again in the next moment. But once you do see it, you will be able to retrace your steps and see it again, even when you lose sight of it.

Does this mean I see it? Actually, friends, I do not. I forgot. I used to know more about than I do now. I followed the reasoning in detail once and saw that it was a game of exponents and how they worked in a round robin of substitution until you arrived back where you started. The ф function was never mentioned way back then in the article I read. It is not the only function that encryption systems have been based on, but I believe it is used in RSA which is the most widely used security system.

If we can make the mathematical mechanics of RSA perfectly and intuitively transparent to ourselves, we have the right to toy around with ideas for an encryption system of our own devising, but not before, certainly not before!

[desiresjab]

Eventually, Yes/No will drag back to the big shoulders of frigid Chicago and give us some good direction and pointed tips. He is a mathematician and computers are his gig. He may have made the encryption process completely transparent to himself already. It is hard telling what a mathematician will get himself up to and involved with. Myself, I am retired so allowed the luxury of going only where I think it matches my desires--one cannot always tell in advanved foreign territory--and my desire is to understand numbers (from natural to complex) on the deepest levels I can.

Instead of prodigy, Paul Erdos called one who remained intellectually active in their dotage a dotagy. I am a dotagy who feels that not only the universe but we ourselves and our consciousness are underlain by deeper layers of unrevealed structure, just like mere numbers are continually rediscovered to be. To us, these deep layers are necessarily more complex, or we would have discovered them first. To me, they possibly hint of what men have correctly or erroneously attributed to objects like spirits and souls in times past. I am one who feels it is probable there are structures of increasing complexity within us that explain age old mysteries and legends, myths, dreams, ESP, prescience, et al. I am trying to edge closer to it before I die. Maybe there is some advantage to be had from pursuing the deeper structures as a pure amateur who will never publish a math paper. I wrote novels, too, that I believe no one will ever read. But in writing them I felt that perhaps I could edge closer to Yeats' golden Byzantium of artifice and creation. What I mean is, I felt that they might have counted in some deeper way as part of another structure, whether any person ever read them or not. Byzantium would recognize me at my death. Okay. Pretty weird.

Back to numbers.

[desiresjab]

The general idea is this: Multiply two gigantic (and I mean gigantic!) primes together, and no one can tell from looking at the result which the hell two numbers you multiplied to get this result.

There are a couple of reasons:

(1) You are in Gigantic territory.
(2) Factoring numbers is inherently hard.

That explains how at the heart of modern e-encryption systems lies the problem of factoring a large number. You--the person trying to break the code--even have some advantages, it would seem at first sight. First, you know it is a composite and not a prime, though it looks exactly like the kind of number computational number theorists would put in great work upon to determine its primality or compositness. Factoring is hard, especailly when your two prime number factors are in the neighborhood of four hundred digits long apiece. You know there are only two and you also know they are relatively close together, i.e. not all that far from the square root of the giant product--and you still cannot do it, even with your computer. If you had a quantum computer, yes, you could do it.

* * * * *

Okay, Jabby boy, you say, how does the ф function come into the picture?

Well, you see, being able to hide what those big factors are is only part of the job, in fact it is the easy part. For anyone can multiply two gigantic primes together and you will not be able to factor them, whether you are the world's best mathematician or the Cray super computer. The clever part was turning the ability to hide those factors into a way to transmit a secure message. This took some footwork worthy of Jersey Joe Walcott. It is like mathematical sleight of hand.

The inventors of RSA truly invented their system, but independently, as they say in the sciences. The work of the guy who got their first, decades before RSA, was immediately marked classified, or his grandchildren might now be vastly rich.

It usually turns out that some smart people got there early in the game, so to speak, before their time.

Anyway, what is needed after multiplying together two extremely large primes, is a way to carry a secure message with the even larger product.

Ask yourself this as you ponder the things above and I go to my slates:

If given the ф of an enormous number but not the number itself, could you determine the number?

If given the ф of an enormous number and given the number as well, could you determine its factors, since ф(pq)=ф(p)·ф(q)?

[desiresjab]

Okay, I think I have a good handle on the basic mechanics of RSA. What I want to do is pare the explanation down to a simple minimum. It will not be the full presentation of how the code is implemented into ASCI, etc., but a bare revealing of the mechanics.

All this is for a later post, once I figure out the best presentation and clear up any lingering questions I myself have.

RSA is indeed clever, but easily understandable with the precise tools we have been using in this thread for months. It is only commutative modular arithmetic in a finite field or ring, to throw some fancy words around. It depends on being able to find the modular inverse of a number with respect to ф(n) when ф(n)=ф(pq)=ф(p)·ф(q) and you only know n but not p or q.

I will try to set it up so it is easy to understand.

[YesNo]

If I remember right, the slash / originated with Fibonacci to represent rational numbers. It took some time to get a decimal representation of them.

Since I am too old to be a prodigy, I'll have to try for a dotagy.

Robert Prechtner uses the ratios of consecutive integers of the Fibonacci sequence to show how this might affect our herding ability as well as other things in nature. I think there is something to this. It is called Elliott Wave theory.

[desiresjab]

If I remember right, the slash / originated with Fibonacci to represent rational numbers. It took some time to get a decimal representation of them.

Since I am too old to be a prodigy, I'll have to try for a dotagy.

Robert Prechtner uses the ratios of consecutive integers of the Fibonacci sequence to show how this might affect our herding ability as well as other things in nature. I think there is something to this. It is called Elliott Wave theory.

That sounds like a likely place for the slash to have begun.

[desiresjab]

Here is my take on RSA encryption. If I am wrong Yes/No or someone else will set me straight.

First, here is the idea in a nutshell. When you raise m to the d power and then to the inverse of d, you end up back at m, the actual message being sent. In this case the inverse is very hard to find because you do not have the right information.

Dusty would like to receive secure messages from Garret, and Mandy would like to know what is in them. So Dusty finds two really large primes p and q, and multiplies them together to get n. Then he calculates ф(n), which he can easily do because he knows what p and q are, and their ф is merely (p-1)(q-1). He then chooses a number d relatively prime to and smaller than ф(n) to use as a power. Then he declares to the world that anyone wanting to send him a message must only raise that message to the power of d first. When Dusty receives a message from Garret it is raised to the power of d, so no one else is able to make sense of it. Dusty can decrypt it because he knows the inverse of d which we call e.

Let the message be m. Garret only wants to send the number 7 to Dusty. He is telling him that afternoon's horse race will be fixed and where to put his money.

Garret makes this number: 7d, and sends it to Dusty. 7d is some other number, very large, certainly it is not 7. All Dusty has to do to know which horse to bet on is this: 7de=7, and we are back at 7, the original message, because d and e are inverses (with respect to ф(n)), and therefore must equal 1 when multipled together as exponents in the term 7de, and of course 71=7. Dusty deciphers the message by simply applying the inverse of what is called the “public exponent,” for that is the power everyone knows they must raise their messages for Dusty to.

Mandy, who is watching all this, cannot figure out what the message is. All she knows is that Dusty raised “some number,” to a very large power, and she even knows the power, which itself is an extraordinarily large number, and she even knows the number n. If she only knew what p and q were she could calculate the value of ф(n), and then she would know how to get the inverse of that huge number d in (7d) with respect to ф(n) that she sees Dusty received. 7de would put her right back at 7, the content of Garret's message to Dusty.

* * * * *

The numbers involved in reality are beyond huge. Depending on the sensitivity of the application, some might have upwards of a thousand digits. To get the ball rolling, encryption folk like the public exponent d to be 65537, but only when that value is relatively prime to ф(n), otherwise they have to choose the next prime number. When Garret raises his message to the 65537th power, the result is a fairly large number, to use an understatement. Mandy cannot raise this number to the inverse power of d with respect to ф(n) because she would need to know the values of p and q to calculate ф(n), which would then make her task easy. Without knowing ф(n), she is reduced to brute force approaches. When the numbers are truly gigantic, brute force can take a long, long time—like the age of the universe or longer—to bring about the correct solution.

We could all agree that 765537 is a pretty large number. It is especially mysterious for Mandy because she does not know the base is 7. So just which number did Garret raise to this large power? You can see where that might take a long time to figure out without the right knowledge which consists of the values of p and q.

That is it. Pretty simple, but it definitely took genius to conceive of.

[desiresjab]

The above post is the bare bones approach. It does not say anything about all the techniques of implementation that require computer experts, or the padding schemes necessary to make encryption more secure, which require even more computer experts. I think one could devise an encryption system without being a computer expert, just from number theoretic knowledge. Let the computer guys figure out how to implement it.

My feeling is that none of the encryption systems are much different from the others in basic technique. They are all based on some common number theoretic function. We have already discovered that elementary number theoretic formulas could all be expressed in terms of one another, if we worked hard enough at it. So, at heart, these systems are not much different from one another, I suspect.

Just remember that the message we sent would be but one alphabetic character or number in a longer message, if we were sending a longer one. We chose a one character message to make the process more transparent.

[desiresjab]

I have finally filled the missing link (for me) in Eisenstein's lattice point proof of Quadratic Reciprocity. It is now clear that the lattice points in the lower left triangle (labeled WAXY in the Wiki-peja article) really do represent quadratic residues. I do not know why it took so long for me to put the last piece in place, but it is now clear. There are so many angles to understand QR from. I understand only one proof of the theorem, but I do fully undrstand that one at least at last. How many ways can you pick pairs from sets of 5 elements and 3 elements with no pair ever being from the same set? Of course the two sets are (p-1)/2 and (q-1)/2.

[Dreamwoven]

All this advanced maths is way above my head, like a new language.

[desiresjab]

All this advanced maths is way above my head, like a new language.

It is above my head, too, DW. That is why I am trying to understand it. A famous mathematician once said math is an unnatural way to think. True but strange, since it seems to be the language of natural things, at the very least their superficial language. For people like Euler and Gauss it is probably not unnatural. For world class mathematicians of any era, I have no idea whether it is a natural way. I suppose it must become so after a while. Even lowly I can notice a difference in my own abilities after having stuck with math assiduously for the last year.

Some of the misconceptions I had along the way and had to correct--why, it's preposterous and laughable. Gauss or Euler or the next level down from them would never have such problems on the same material, I imagine. They invented half the stuff I am still trying to understand well. All they had to guide them were their own instincts.

* * * * *

Anyone with a cosmological thought should voice them here. This is not a math thread but only seems so at the moment. Thoughts on the subject that are not math-oriented are not interrupting anything but adding to the discussion's breadth. I do not know peoples' ages on here. I know death begins to preoccupy the mind after a certain age. Death and cosmology go together.

* * * * *

Certain questions from months ago are still haunting me. Are types of universes possible which are logically impossible to us? I don't know how one would ever answer that.

[desiresjab]

I see now how e, pi and i got into those number theory equations. Like Yes/No said they are for finding the roots of unity for any xn-1=0, a long standing problem historically. I found an abstract algebra video where the instructor explained it with the unit circle for the roots of uinity of x5-1=0. He showed how to get roots of unity every 72 degrees, by multiples of one initial position equation.

They had me wondering, too. That is why I italicized them. Roots of unity are slightly different than the roots of equations we studied in high school, where we would surely be able to find two roots to x5-1=0, but what about the other three, for by the Fundamental theorem of algebra it should have five roots? Yes, even lowly 1 has three other roots. And the above is a way to get them.

I also straightened out my confusion on the use of the word Order. Group theory uses the term two different ways:

1 The number of elements in the group
2 The lowest power to which a group element has to be raised to return to the identity value.

The latter is actually what they officially named the Carmichael function over in number theory. The latter also roughly equates to the order of the element of a modulus ring (which we are so familiar with by now) over in number theory.

I will just chip away, concept by concept--here on groups, here on rings, here on fields, here on algebraic numbers, here on forms, here on categories--until my Frankensteinian collage starts to take form. I do not have the background to forge straight ahead. I do have a fair instinct for where to chip to get what I want, and Yes/No seldom points the right way with the back of his neck.

[Dreamwoven]

I wouldn't call anything "unnatural" Maths is just very difficult to wrap my mind round, is all, I don't have the education for it and find it too difficult to learn, though I am sure it can be learned. The Wikipedia item on cosmology expresses it well: https://en.wikipedia.org/wiki/Cosmology.


It is above my head, too, DW. That is why I am trying to understand it. A famous mathematician once said math is an unnatural way to think. True but strange, since it seems to be the language of natural things, at the very least their superficial language. For people like Euler and Gauss it is probably not unnatural. For world class mathematicians of any era, I have no idea whether it is a natural way. I suppose it must become so after a while. Even lowly I can notice a difference in my own abilities after having stuck with math assiduously for the last year.

Some of the misconceptions I had along the way and had to correct--why, it's preposterous and laughable. Gauss or Euler or the next level down from them would never have such problems on the same material, I imagine. They invented half the stuff I am still trying to understand well. All they had to guide them were their own instincts.

* * * * *

Anyone with a cosmological thought should voice them here. This is not a math thread but only seems so at the moment. Thoughts on the subject that are not math-oriented are not interrupting anything but adding to the discussion's breadth. I do not know peoples' ages on here. I know death begins to preoccupy the mind after a certain age. Death and cosmology go together.

* * * * *

Certain questions from months ago are still haunting me. Are types of universes possible which are logically impossible to us? I don't know how one would ever answer that.

[desiresjab]

I remember reading a book on Teleology back in my early twenties. No, it was called The Cosmological Arguments, and had a section called the Teleological Argument. I think it also had a section called The Argument From Design. I gave philosophy the college try in those days, wading through many traditional names. It was hard wading. I did not understand a lot of it. I found Heidegger hardest of all with also the hardest name to remember how to spell. What amazed me was how men could have so many thoughts and arrive at so many fast convictions about the universe. I was wondering if I would ever arrive at one fast conviction about the joint. Lo and behold, over the last few years I finally did. You have probably heard it. Whatever there is--universe or multiverse--it cannot have come from nothing. If something exists now, this so-called nothing we came from was a false nothing, for it at least contained the possibility, the potential for something to come about, and potential is not nothing. The only thing that can come of nothing is nothing.

That does not mean that it was my thought, only that I was led there as a result of my own reflections (or so it seems) rather than lifting it from someone. That said, a million people must have said it. I said and I feel it. It is one of my few hardcore convictions about cosmology.

[desiresjab]

I will soon be off for a few days again, traveling to visit an ancient parent who is still fully sentient. I never take a computer. My brain is on its own when I travel, which forces me back to pen and paper if I need to calculate once I arrive, for I always drive. Driving at night on lonely highways is great for deep thinking. Just make sure no elk clips your mirror off and smashes your windshield, i.e. enter those fog banks at a crawl, honk your horn to be safer (it's lonely, right?) because elk may panic or be blinded and run right toward your headlights. They forgot already they are on a road, if they ever remembered. The noise scares them out of your way, hopefully.

* * * * *

I do not want to go away leaving misconceptions. The lattice points in WAXY in Eisenstein's rectangle in the Wiki-peja article on QR proofs represent all the ways that sets of 5 and of 3 elements can combine in pairs. Each pair is not interesting in itself, but only for which triangle it lies in, each point does not map back somewhere that tells us anything. Their whole point is that there are precisely this many of them, i.e. precisely φ/4 of them. Eisenstein never mentions φ, but what his four smaller rectangles are doing is dividing: φ(pq)/4, which can use the Euler phi function for prime rectangles, at least. Of course, he is working from the angle of Euler's criterion, because he has to make that connection and show that his diagram actually represents quadratic residues, at least in their correct number. Once we have understood him we can take the shortcut of φ/4 in perfect safety. It will work every time. However, it is faster just to multiply, but nice to know this other function we claim is ubitquitous (the phi function) offers us yet another example.

The diagonal should not be taken as another division by 2, as I mistakenly did some months past. It is a simple scalar, which can be a confusing word in mathematics. The diagram takes a ratio via the diagonal, nothing more.

I am now interested again in the idea of that limiting ratio. In other words, how much relative difference between p and q will guarantee a different number of points in the two triangles of WAXY when the Legendre symbol of the two primes is even, for we realize two 4n+3 primes must always have a different number of points, regardless of their difference or ratio. With such a small slope for its diagonal we do not expect the pair (5, 41) to produce equal numbers of points, for instance. I believe I calculated them and there were 24 and 16 points in the two triangles respectively. On the other hand we know quite well (without having actually proved it) that twin primes will always have the same number of points in both triangles. We have seen it in primes that are not twins, too. This leads us to wander what the limiting ratio is. Probably an easy question. But few questions in math are easy for me.

So anyway, that is one of the things I will be thinking about in the car, all warm and lonely. I just could not live without lonliness, or maybe would not care to anymore. It seems like a real gift bestowed through cruelty sometimes. If life ends forever, I believe that would be cruel if a conscious creator had at least as much empathy as us, and I know the thought of final ending is cruel. I think we would hope for a conscious creator with a fair bit more empathy than ourselves. Minus a conscious creator, let us hope the structure of our being is so complex and deep that it happens to include an afterlife of some kind. We understand four percent of the "stuff" in the universe so far. We have probably not penetrated our own structure even that deeply yet.

[YesNo]

If the limit does not converge to one number there may be a set of numbers it converges to that would be interesting.

I agree that nothing comes from nothing. The way I look at it is that before the beginning there was no unconscious matter. There was nothing. After the beginning there must still be no unconscious matter. What we think is unconscious matter is an illusion.

Part of the problem with e raised to the pi times i equaling -1 is that we think of ex as a function graphed on an x and f(x) plane and it increases exponentially. But in that case the x is always real. I used to confuse that x-f(x) plane with the complex plane, but it is different. If one goes pi radians about a unit circle one is at the -1 point a 180 degree turn or a pi radian turn.

Math may be a deceptive way to think about reality, more than an unnatural way to think. It leads one to think that determinism and randomness are to be expected, but I don't think there is anything that is deterministic or random in the universe unless we construct it to be so, like a mathematical theory or a computer (which eventually breaks down destroying the determinism we put into the machine).

I think there are many universes since a single universe cannot be infinite without destroying the possibility of life, but they are all the same. They would follow an evolution that is similar to a spiral rather than a circle.

[desiresjab]

If the limit does not converge to one number there may be a set of numbers it converges to that would be interesting.

I agree that nothing comes from nothing. The way I look at it is that before the beginning there was no unconscious matter. There was nothing. After the beginning there must still be no unconscious matter. What we think is unconscious matter is an illusion.

Part of the problem with e raised to the pi times i equaling -1 is that we think of ex as a function graphed on an x and f(x) plane and it increases exponentially. But in that case the x is always real. I used to confuse that x-f(x) plane with the complex plane, but it is different. If one goes pi radians about a unit circle one is at the -1 point a 180 degree turn or a pi radian turn.

Math may be a deceptive way to think about reality, more than an unnatural way to think. It leads one to think that determinism and randomness are to be expected, but I don't think there is anything that is deterministic or random in the universe unless we construct it to be so, like a mathematical theory or a computer (which eventually breaks down destroying the determinism we put into the machine).

I think there are many universes since a single universe cannot be infinite without destroying the possibility of life, but they are all the same. They would follow an evolution that is similar to a spiral rather than a circle.

Hello. Happy to be back.

I have noticed that opinions hardly interest me anymore. Even my own cloy my thinking. I am gorged. To me an understanding of how Carmichael numbers can form ideals is worth any number of opinions or unapproachable speculations right now. I have also been wont to speculate toward much larger pictures, and will be so again. But for now my speculations must end in mathematical truth I know is there. Full understanding of consciousness is not even remotely possible at this time. But for the assiduous, Carmichael numbers are, unless you happen to plum run out of brains. I have my hopes up that I have not run out yet.

While I was away I vaguely remembered one statement from an article I read that every Carmichael number that is not already a double of a Carmichael number will have a double which is. I may have misread that statement. But I began to wonder anyway if each Carmichael and its single double form their own ideal set of which other Carmichael numbers are not members and have nothing to do with. I do not know yet. Haven't even checked yet. I was hoping you knew. I am looking for a Trojan horse in this seige.

Either Dedekind wins or I win. It has always been that way once I become obsessed. Directed obsession is the best tool a person has, it is the best one I have found. People set limits for themselves that are not necessarily true. It is a natural habit. I am interested in my own limits, the real ones, if ultimately there are any. I do not want to do IQ tests or physical puzzles, I want to see how far I can penetrate the nature of numbers. Seeing into numbers is seeing into the universe and maybe into God. Writers have a connection too, for God just spoke everything into existence according to one old text. I take the Byzantian creative ideal of Yeats as seriously as the math ideals of Dedekind. When Billy said, "The best lack all conviction, while the worst are full of passionate intensity," I do not think he meant my intensity, or his own, which aimed at understanding. Math is rarely "Modified in the guts of the living," however, as another Billy almost said, which shows the difference in the arts. Euler was as creative as Yeats. I nestle up to both.

Lunch is over. I hear the battle horn. The seige resumes.

[desiresjab]

It is not all n-1 residue classes that are false witnesses to make a Carmichael number. Only those that are relatively prime to n. In the case of a Carmichael number, which are squarefree, one would have to get a factor for Fermat's criterion to be accurate, but it wouldn't be accurate for an = a (mod n). That would still work.

Consider 561 = 3*11*17, a Carmichael number (assuming the python is correctly programmed):

3561 = 3 mod 561, but 3560 = 375 mod 561
11561 = 11 mod 561, but 11560 = 154 mod 361
17561 = 17 mod 561, but 17560 = 34 mod 361

There are seem to be at least three layers of tests each restricting the exponent of the witness a bit more:

an = a mod n
an-1 = 1 mod n Fermat's test
a(n-1)/2 = (a/n) mod n Euler's test

I assumed what was in red after you said it, but doesn't the statement below contradict it from the first paragraph of the linked article?

"There are composite numbers n which fail this test no matter how we choose a...."

http://www.sciencedirect.com/science/article/pii/S0022314X07002089

[YesNo]

As far as the best lacking conviction goes, it sounds like whining. But no one quotes me like they quote Yeats, which is probably a good thing.

I don't know much about Carmichael numbers except what I have explored with you. So Carmichael numbers come in pairs based on the doubling idea. That might make sense because phi(2) = 1. So Carmichael numbers can be even.

Edit: There are three tests. If one uses an = a mod n then all integers a will give the desired result for a Carmichael number a. However, if one uses an-1 = 1 mod n, then a has to be relatively prime to n for that to work. The third test should not have any Carmichael numbers although there are pseudoprimes. At least that is how I see it at the moment. I might be wrong.

[desiresjab]

Very good. I think you have it.

But wait! I have to have misread. How can a Carmichael number have a double when they are all odd? Scratch that bad idea. Which means I am back to wondering how something that is not a multiple can be an ideal. There was in the beginnings an ideal number theory, then there came a more general theory of ideals, which is how Carmichaels were made into ideals. At that point I believe they are talking about algebraic integers and particular behaviors in groups instead of everyday integers. The wider theory still looks like Fermat's Little Theorem with different representatives as the exponents which mean the same thing for groups as the familiar exponents mean for numbers. I will try to zero in on those group behaviors.

[desiresjab]

I was pleased and surprised to learn that in the ring of Polynomials all equations whose constant eqauls zero form an ideal because they all tend to zero as x vanishes toward mathematical nothingness.

Another fascinating fact was that in a ring of polynomials Carmichael numbers with only two factors occur. These are calleg Gauss-Carmichaels, or just G-Carmichaels. And I believe that left and right multiplication simply refers to the non-commutative nature of multiplication in some rings and groups. I now think when a prime splits it probably refers to the unavoidability of non commutative multiplication in the attempt to factor into prime ideals. The subtle differences between groups and numbers have to be observed. A ring of integers is not a ring of polynomials. It seems there are serious differences and serious similarities.

I think I read that the ideal also gives pros a reliable measure of how far from completely factorable a polynomial is in ideals. They add some exponents to get this value. I forget exactly which exponents though. A line of research begun by old Gauss two hundred years ago, I believe. It is ahead of myself and solid understanding. I often have to read ahead of myself.

P.S. Something like they add all the exponents not forced to zero when they impose conditions I forget. I cannot even remember the precise situation. Sometimes these half-memories do not work out, like the idea of a Carmichael double. I think I may have confused that with some kind of ideal double which indeed might exist.

[YesNo]

It is interesting that Gauss-Carmichaels can have as few as two factors. I remember a few days ago being convinced that there has to be three factors in a Carmichael number in the integers. This makes me wonder why factoring in a ring of polynomials can generate something different.

[desiresjab]

It is interesting that Gauss-Carmichaels can have as few as two factors. I remember a few days ago being convinced that there has to be three factors in a Carmichael number in the integers. This makes me wonder why factoring in a ring of polynomials can generate something different.

It makes me wonder too. Maybe one has to get down among the greasy gears and watch this style of factorization for a while. I have never seen it done. I assume it is something different from the factoring one does in high school. How does it work? Can you factor such an expression for me?

[YesNo]

Just to make sure I am not confused I looked at this article: https://en.wikipedia.org/wiki/Polynomial_ring

A polynomial ring needs some symbol X and coefficients in some ring such as the integers. Then (x+1)(x+2) = x2+ 3x + 2 is a factorization. I don't think it is anything more than that, but again, I might be missing something.

[desiresjab]

Just to make sure I am not confused I looked at this article: https://en.wikipedia.org/wiki/Polynomial_ring

A polynomial ring needs some symbol X and coefficients in some ring such as the integers. Then (x+1)(x+2) = x2+ 3x + 2 is a factorization. I don't think it is anything more than that, but again, I might be missing something.

I am trying to determine if that is precisely what they mean when they call a factor irreducible in a polynomial ring. It know it equates somehow to primes but is not quite a prime. I also suspect it perhaps has something to do with the roots of unity technique I observed on the abstract algebra video.

For instance, it seems to me that your example equation would not be further reducible because, for one criterion, it has already been factored to linear terms. I believe that is a strong criterion for an irreducible factor--that it be in linear terms. Even something like x4+1=0 has to be in linear terms to be factored irreducibly, I believe. I don't know if all equations can be. In fact, though, I may have read that some cannot be made irreducible even over the Complex numbers. There is much to learn along the way.

More intense study of groups has to be next. In the Wiki-peja article on Group Theory it states that Algebraic Number Theory is a special case of Group Theory, so follows the rules of the latter. This looks like Plymouth Rock to me. It means that to get my chokehold on ideals I will next have to retreat more intensely at Group Theory. I am out of sequence in my studies, but maybe I can survive it. My Abstract Algebra is ahead of my Linear Algebra (practically non existent for me) and Group theory (medicocre to half-assed decent). I seem to be learning them all at the same time, but only because it seems necessary in the quest to nail down ideals in polynimial rings.

[YesNo]

That example should be completely factored. This Wikipedia article provides a good review: https://en.wikipedia.org/wiki/Irreducible_polynomial

Whether a factorization of a polynomial is reducible or irreducible depends on the ring the coefficients belong to.

There a distinction between primes and irreducibles, however, they are the same if one is in a "unique factorization domain" like the integers. This is what makes unique factorization important and not obvious even though it seems obvious. All of these distinctions can be either a pain in the rear or a delightful puzzle that one cannot put down.

Edit: In addition to not being able to be factored further, a prime p has the property that if p divides n = ab, then either p divides a or p divides b. If a natural number n could be factored into irreducible positive integers in more than one way, that would not be the case. I wonder if it is proper to define a prime by this property or to use this property to define a unique factorization domain?

[desiresjab]

A fellow before my last sleep was discussing in his advanced mathematics video how computers will change formal mathematics. He says some of the infinite operators in math of the past few centuries will be defined differently in this century in finite terms that can be computed. You canot compute something requiring infinite operations to get the job done.

This fellow calculated the roots of unity of x15-1 without any recourse to problematical Complex numbers which will not factor nicely. He makes use of something he calls Quadrance. He does not want to use its usual name of Norm because that is his name and Norm is used in another part of mathematics with a completely different meaning.

It appears mathematicians have been troubled over The Fundamental Theorem of Algebra since Gauss came up with it, kind of like Euclid's parallel postulated bothered great mathematician's for centuries. It turns out they had good reason to be bothered, since the postulate is not universally true. I wish I were going to be around to observe the eventual fate of The Fundamental Theorem of Algebra.

According to this fellow, ideals went a great ways toward eliminating some of the conflicts in the field by allowing their own arithemetic in which complete factoring was accessible. I will have to make sure of that last statement.

[YesNo]

Interesting observation about the Fundamental Theorem of Alegebra. I'm trying to see how it is like Euclid's parallel postulate, but it may be.

What is the link to the video about the change away from infinite operators in formal mathematics?

[desiresjab]

Interesting observation about the Fundamental Theorem of Alegebra. I'm trying to see how it is like Euclid's parallel postulate, but it may be.

What is the link to the video about the change away from infinite operators in formal mathematics?

Actually, he did not use the word operator, that was my bad handoff, he said away from infinite processes. I like this guy. He knows the mathematics really well, well enough to have deep convictions about the way math should be fundamentally reorganized and taught. He is a real nut in a way, a Kronecker throwback who even teaches something called Rational Trigonometry and eschews most theories in mathematics that smack of the infinite. He at least tries to make it smack less. Out of all the math lecturers I have watched on the internet he is the best, most organized, easy to see, hear and understand with the best laid down presentation. He has a ton of videos and he got me hooked. I can lead you to the right YouTube menu, and maybe the right video.

In the integers Z all ideals are principle ideals, because all ideals are a multiple of the unit. Sometimes it is hard to distinguish if they mean algebraic integers when they say integers. If integers have only this multiple type of ideal, then it must be the algebraic integers in complex numbers that allow other types of ideals. Anyway...

https://www.youtube.com/watch?v=H8xBlLWdzBE

https://www.youtube.com/watch?v=GMZoXXaOFeQ

If it was neither of those it was probably in one of his two videos on Galois theory, though later on he talks about it more extensively.

[YesNo]

I think he mentioned it in the first of those two lectures. He does not believe in infinite sets nor in processes that involve an infinite number of operations. The Axiom of Choice would be needed to assume one could do an infinite number of steps or choices.

I noticed in his two lectures that he called things "prime" that I would have only called "irreducible" and not prime. He did provide an example of a ring without unique factorization in the second lecture. It is good to know that such things exist because it makes one value the Euclidean algorithm for division.

I wonder why he does not like having infinite sets? In my case, I do not assume that the universe has to contain infinitely many integers for this to be true. I wonder if there is more to his objections than that.

[desiresjab]

I think he mentioned it in the first of those two lectures. He does not believe in infinite sets nor in processes that involve an infinite number of operations. The Axiom of Choice would be needed to assume one could do an infinite number of steps or choices.

I noticed in his two lectures that he called things "prime" that I would have only called "irreducible" and not prime. He did provide an example of a ring without unique factorization in the second lecture. It is good to know that such things exist because it makes one value the Euclidean algorithm for division.

I wonder why he does not like having infinite sets? In my case, I do not assume that the universe has to contain infinitely many integers for this to be true. I wonder if there is more to his objections than that.

He talks about his objections more extensively in other videos. He feels that mathematics has gone the way of complex convolutions hardly anyone can understand. I only partially agree. But if there is an easier way to approach many topics as he claims, I am interested. His claim that e and i and pi are not needed to find the roots of unity of x15-1, in other words cyclotomic monic polynomials, is quite appealing. He does it on the video. I only partially followed the reasoning, but I will go back for another helping.

Something he complains heavily about is the lack of good examples in these highly abstract areas such as infinite sets. Though he does a fine job I could level the same complaint his way. Where is my example in detail of an ideal other than the simple principle ones of the integers? After extensive investigation I still have no idea how a Carmicheal number can have anything to do with an ideal. A third grader could understand a principle ideal, now show me how these other objects can be ideals when they are not multiples of a generator, or whatever. I am a little peeved at the math industry myself, you might say. I have put in enough effort. Are these idiots lost in their abstractions hiding the answer or am I simply too thick? Certainly it takes time, for there are times I will read right over a key statement multiple times without noticing it holds the answer. I do not know what I am overlooking this time.

[desiresjab]

Continue with my own whines...

It was very neat, for instance, how they "said," (and that is about all they did) all equations that have a constant of 0 form an ideal. Okay. Neat. How is that? Are they multiples of one another? I think not. Then in exactly what way does having a constant of 0 unite them into something we call an ideal? I don't get the parameters. What qualifies to make something an ideal, and what does not qualify, for that matter?

[YesNo]

I might be misunderstanding this, but regarding polynomials that have a constant term 0, they would be generated by the polynomial x. Take any polynomial p out of the ring of polynomials, say p = anxn+...+a1x + a0 and multiply p by x. You will get another polynomial that has the constant term 0 because the polynomial p from the ring, which might have had a constant term a0 not equal to 0, was multiplied by x making the constant term of the product, xp, equal to 0.

I don't see, at the moment, how Carmichael numbers relate to ideals, but I suspect they are.

[desiresjab]

I might be misunderstanding this, but regarding polynomials that have a constant term 0, they would be generated by the polynomial x. Take any polynomial p out of the ring of polynomials, say p = anxn+...+a1x + a0 and multiply p by x. You will get another polynomial that has the constant term 0 because the polynomial p from the ring, which might have had a constant term a0 not equal to 0, was multiplied by x making the constant term of the product, xp, equal to 0.

I don't see, at the moment, how Carmichael numbers relate to ideals, but I suspect they are.

So the constant term of x is zero but we multiply by it as if it were there anyway. It is there (invisibly) in the constant term which has to be multiplied by every term in p, so that particular row of the polynomial multiplication will be all zeros. Is this correct?

If I am not misinterpreting this, you are saying that every polynomial without a zero constant can be multiplied by any polynomial that has a zero constant and the result will be a polynomial that has a constant of zero. Is this correct?

But would not this operation put any polynomial at all in the ring?

And couldn't any polynomial with zero constant force every polynomial into its particular ring?

[YesNo]

I assume we want a set, an ideal, that contains all the polynomials with 0 in the last constant term. One way to get that would be to take all the polynomials and multiply them by x + 0. Some polynomials would not be in this ideal. For example the polynomial x + 1 would not be in the ideal. It has 1 as a constant term. But we could use it to get (x + 0)(x + 1) = x2 + x + 0 which is in the ideal.

If we used x2 + 0 as the generator we would miss x + 0 in that ideal. Not all polynomials with 0 in the constant term would be in that ideal.

[desiresjab]

I assume we want a set, an ideal, that contains all the polynomials with 0 in the last constant term. One way to get that would be to take all the polynomials and multiply them by x + 0. Some polynomials would not be in this ideal. For example the polynomial x + 1 would not be in the ideal. It has 1 as a constant term. But we could use it to get (x + 0)(x + 1) = x2 + x + 0 which is in the ideal.

If we used x2 + 0 as the generator we would miss x + 0 in that ideal. Not all polynomials with 0 in the constant term would be in that ideal.

Yes, I am finally beginning to see. Any equation with 0 as a constant will generate from any other equation a multiple of itself. It can generate every multiple of itself on any other equation, but not the same multiples that most other 0 constant equations can generate of themselves, for they are different multipliers unless their difference is a trivial one of not being reduced fully.

Now, just what is the supposed infinite set here, the multiples of itself that a 0 equation can generate from one other equation, or what it can do to all equations? It must be the latter from what you say. Any equation with 0 as constant will spread that effect on multiplication. I am hesitant to use the word class right now because I know that word is probably used in a precise manner in the language of ideals ahead.

So, from all other equations it generates an infinite set by multiplying itself with them. This is the infinite class of ideals associated with this particular equation. Now another distinct equation with 0 as a constant does the same thing to all other equations, and thereby generates its own infinite class of ideals based on being a multiple of itself. That is how I see it at the moment. Next I need to see how Carmichael numbers form ideals, unless I have still got something wrong. I am almost there.

[YesNo]

The infinite set would be the set of polynomials that have the generator as a factor.

If one uses 0 as the generator one would get the zero ideal which has only 0 in it. The ideal with 1 as the generator would be the opposite extreme and contain all the polynomials.

It seems to me at the moment that when one looks at ideals one is looking at all the multiples of a generator. Often I think of a prime p in the integers as a positive integer such that if p divides ab then p divides a or p divides b. That would be looking at a division property of the prime rather than all of its multiples. For a prime ideal P one would rewrite that as if ab are in P then either a is in P or b is in P. https://en.wikipedia.org/wiki/Prime_ideal

[desiresjab]

In other words the generator only makes a single copy of each equation it comes in contact with. If I have a generator equation G and another equation F, then multiplicatiobn produces a multiple of G. It is not exactly a multiple of F, because F did not have 0 as constant and the new equation does. If G multiplied itself with the F more than once it would in effect be squaring itself, which is not part of the deal.

So a generator function acts once upon each equation in the world without a zero constant, and thereby produces an infinite set of equations with 0 as constant because there are infinite equations without 0 as constant to multiply itself with.

To me this is different than a single integer generating an infinite set of intervals (ideals) on the number line, for I see nothing regularly spaced about these new equations generated by G. However, I have certainly read that every ideal in integers or Gaussian integers is a principle ideal. As far as I can tell, this is not true of algebraic integers, which are strictly the roots of equations. Every Gaussian integer is not the root of some equation, is it? There is some confusion still whether expositors are speaking of Gaussian integers or algebraic integers at a given time in a discussion. That is, algebraic integers would not entirely fill the lattice points of the Complex plane as Gaussian integers do. Is there any truth to this or am I misinterpreting something?

[desiresjab]

If I took this new equation generated by GxF and multiplied it times each integer, then I would would have infinite multiples of G from one F, just as I can generate with a plain integer, but that does not seem to be part of the definition (the deal). Here the equations (F) without 0 as constant stand in an infinite line waiting to be multiplied with G. Those single multiplications on each F generate the infinite set of ideals (equations which are all multiples of G).

Name a new multiple of G as H. Now G would divide each H it generates. But each H in turn does not have to divide every other H, anymore than 6 has to divide 9 just because they are both multuiples of 3. Though not multiples (necessarily) of one another, some of the new equations in the infinite set would have to be multiples of some other equations in the new set, just as 6 is not a multiple of 9 but is a multiple of 12. Equations in the infinite set which happen to be exact multiples of one another would form an infinite subset within the set. Whether this subset carries much meaning I do not know.

[desiresjab]

In the vector-ball diagram in the Wiki-peja article you linked to, am I to take it that the top purple row is actually an infinte row of pure primes? Otherwise they would be saying that only 2, 3, and 5 can generate ideals.

[desiresjab]

Broken back to English, it seems to go like this for ideals in commutative rings:

1 Prime ideals are of the form nZ, where n is a prime.

2 Primary ideals are composed of powers only of a prime element. This means n, the prime element, is also primary, even with the lowly power of 1.

3 Semiprime ideals are combinations of more than one prime, but which are also square-free.

I heartily agree with Wildberger that more good examples are needed. The Wiki-pega article on Semiprimes is valuable just because it gives a specific example, which cuts off a lot of questions at the pass. The article notes with the required specificity for dummies that 30Z would be a semiprime ideal, where as 12Z would not be. Mathematicians act like specific examples are going to kill them or lower their princely standards. The example makes it clear that 30 is semiprime because its factors are no more than single powers of primes.

For myself, adjusting to the language of ideals will take further familiaization to become entirely comfortable. Experts often talk somewhat loosely among themselvs, and tend to continue this trend in their expositions. Most mathematicians are poor expositors when it comes to bringing their abstract notions out of the darkness for laymen or even interested amateurs.

There is a reason for this: the second job is more formidable. With another expert, talking over concepts is easy. As Wildberger notes somewhere, it is basically Santa Claus to the Easter Bunny power, a pure manipulation of symbols. When I am done, I will be able to make the notion of ideals and their ramifications clear to an interested person. If it is clear to me, I should be able to do that.

By the way, I am looking at 12Z. It is not a prime, it is not primary, and it is not semiprime either. It is white on the diagram. It must play the role of a strict composite in ideals. I don't know, I am just guessing. I will overcome many impasses and wrong notions as I continue to chip away. In the end I have to be able to perform the arithmetic of ideals as easily as I can perform modular arithmetic in integers.

[YesNo]

In other words the generator only makes a single copy of each equation it comes in contact with. If I have a generator equation G and another equation F, then multiplicatiobn produces a multiple of G. It is not exactly a multiple of F, because F did not have 0 as constant and the new equation does. If G multiplied itself with the F more than once it would in effect be squaring itself, which is not part of the deal.

An ideal is a subset of a ring. If G is a generator of an ideal and F is any element in the ring then GF is in the ideal although F may not be in the ideal. In particular since G is in the ring, then GG is in the ideal. So the square of G is in the ideal, just as one would expect the square of any prime to be in the set of multiples of that prime in the integers.



So a generator function acts once upon each equation in the world without a zero constant, and thereby produces an infinite set of equations with 0 as constant because there are infinite equations without 0 as constant to multiply itself with.

To me this is different than a single integer generating an infinite set of intervals (ideals) on the number line, for I see nothing regularly spaced about these new equations generated by G. However, I have certainly read that every ideal in integers or Gaussian integers is a principle ideal. As far as I can tell, this is not true of algebraic integers, which are strictly the roots of equations. Every Gaussian integer is not the root of some equation, is it? There is some confusion still whether expositors are speaking of Gaussian integers or algebraic integers at a given time in a discussion. That is, algebraic integers would not entirely fill the lattice points of the Complex plane as Gaussian integers do. Is there any truth to this or am I misinterpreting something?

I think every Gaussian integer would be the root of a polynomial with integer coefficients. Let a + bi be a Gaussian integer, where a and b are regular integers. Note that (a + bi)(a - bi) = a2 + b2, an integer. Multiply together (x - (a + bi)(x - (a - bi)) to see if this forms a polynomial with integer coefficients. I get x2 - 2ax + a2 + b2, unless I made a mistake. So the arbitrary Gaussian integer a + bi is the root of a polynomial with integer coefficients.

A principal ideal is an ideal generated by a single element: https://en.wikipedia.org/wiki/Principal_ideal

I am asking myself if it is true that every ideal in the Gaussian integers is a principal ideal. I think it is, because of what you mentioned, but I will have to find a proof.

[YesNo]

In the vector-ball diagram in the Wiki-peja article you linked to, am I to take it that the top purple row is actually an infinte row of pure primes? Otherwise they would be saying that only 2, 3, and 5 can generate ideals.

Yes, the diagram was only partial. The purple row contains all the primes in the full diagram which can't be written out. However, now that you pointed it out, that diagram mentions a bunch of terms: prime ideals, semi-prime ideals and primary ideals. They apparently mean different things and have some use value, but now I am trying to clarify in my own mind what those are.

Edit: I just saw your recent post. I think you have clarified these terms.

[desiresjab]

Yes, the diagram was only partial. The purple row contains all the primes in the full diagram which can't be written out. However, now that you pointed it out, that diagram mentions a bunch of terms: prime ideals, semi-prime ideals and primary ideals. They apparently mean different things and have some use value, but now I am trying to clarify in my own mind what those are.

Edit: I just saw your recent post. I think you have clarified these terms.

If a principal ideal is generated by a single element, then primary ideals must also be principal ideals, since they are powers of a single element. Keeping all the lingo straight in order to go farther apparently comes with the territory. Specific examples are the color gold in an otherwise black and white setting. All prime ideals are primary, but obivously all primaries are not prime. I still do not know the official classification of 12Z.

Primes are both semiprime and primary, two different branches, but which makes sense because they are the geneators of everything after all (not sure about 0, however), so they should generate all the branches, it seems intuitively.

* * * * *

It has diverted my attention whether all integers defy the Fundamental Theorem of Arithmetic over the Complex. I saw 5 factored two different ways. The same technique should work for any prime--just use the conjugate. Since a composite can be broken into prime factors (which themselves defy unique factorization), then the composite has more than one factorization as well. The number of ways to combinatorially compute the division algorithm would be a simple extension of adding in more factors, but some of which do not work together. Hence, all integers defy unique factorization over the Complex field.

Excuse that little aside. I needed that. I am that rusty in areas.

[YesNo]

I was thinking about the different terms today as well while walking. This is how I see it.

If one has a field, a special kind of ring where all elements (except 0) have a multiplicative inverse, then there are only two trivial ideals: the whole ring and the set containing only 0. We can forget about fields except as sources of examples.

So a ring has to have elements that do not have multiplicative inverses for ideals to be interesting. The integers would be an example of such a ring as well as polynomials.

Here are the definitions:
1) Ideal, a subset of a ring generated by a finite set of elements.
2) Principal Ideal: an ideal that can have the set of generators reduced to one element.
3) Zero ideal: the ideal generated by the 0 element and containing only 0.
4) Unit ideal: the whole ring generated by a unit such as 1.
5) Prime ideal: a principal idea generated by an element p such that if ab is in the ideal then either a is in the ideal or b is in the ideal. For example, the ideal generated by 6 would not a prime ideal since 36 = 4 * 9, but neither 4 nor 9 are multiples of 6 and so they are not in the ideal. The ideal generated by 6 would not be a prime ideal, as expected.
6) Semi-prime ideal (radical ideal): is an ideal generated by a square-free integer. Here the ideal generated by 6 = 2 * 3 would be example and the ideal generated by 12 would not be an example.
7) Primary ideal: is an ideal generated by the power of a prime.

I am sure there are other critical definitions and then one needs to find out how these work in many different rings.

In the Gaussian integers 5 is not a prime because it can be factored since 5 = 12 + 22 = (1 + 2i)(1 - 2i). This is true of all primes in the regular integers that are congruent to 1 mod 4. But primes in the regular integers that are congruent to 3 mod 4 cannot be represented as a sum of squares and so they are prime even in the Gaussian integers since they are irreducible. The Gaussian integers are supposed to be a unique factorization domain which means irreducibles are primes. To construct the Gaussian integers add to the regular integers i = sqrt(-1). The example that did not have unique factorization was when one added sqrt(-5) to the regular integers. The unique factorization failed in that case, but these are not the Gaussian integers.

[desiresjab]

I was thinking about the different terms today as well while walking. This is how I see it.

If one has a field, a special kind of ring where all elements (except 0) have a multiplicative inverse, then there are only two trivial ideals: the whole ring and the set containing only 0. We can forget about fields except as sources of examples.

So a ring has to have elements that do not have multiplicative inverses for ideals to be interesting. The integers would be an example of such a ring as well as polynomials.

Here are the definitions:
1) Ideal, a subset of a ring generated by a finite set of elements.
2) Principal Ideal: an ideal that can have the set of generators reduced to one element.
3) Zero ideal: the ideal generated by the 0 element and containing only 0.
4) Unit ideal: the whole ring generated by a unit such as 1.
5) Prime ideal: a principal idea generated by an element p such that if ab is in the ideal then either a is in the ideal or b is in the ideal. For example, the ideal generated by 6 would not a prime ideal since 36 = 4 * 9, but neither 4 nor 9 are multiples of 6 and so they are not in the ideal. The ideal generated by 6 would not be a prime ideal, as expected.
6) Semi-prime ideal (radical ideal): is an ideal generated by a square-free integer. Here the ideal generated by 6 = 2 * 3 would be example and the ideal generated by 12 would not be an example.
7) Primary ideal: is an ideal generated by the power of a prime.

I am sure there are other critical definitions and then one needs to find out how these work in many different rings.

In the Gaussian integers 5 is not a prime because it can be factored since 5 = 12 + 22 = (1 + 2i)(1 - 2i). This is true of all primes in the regular integers that are congruent to 1 mod 4. But primes in the regular integers that are congruent to 3 mod 4 cannot be represented as a sum of squares and so they are prime even in the Gaussian integers since they are irreducible. The Gaussian integers are supposed to be a unique factorization domain which means irreducibles are primes. To construct the Gaussian integers add to the regular integers i = sqrt(-1). The example that did not have unique factorization was when one added sqrt(-5) to the regular integers. The unique factorization failed in that case, but these are not the Gaussian integers.

I pretty much have most of that. But I forgot something critical, which I was supposed to know, which I put in blue; and I have completely overlooked something critical, if it is true, which I have put in red.

[desiresjab]

Then again, it seems like I have done exactly the same thing to each of these:

Factorization of 5.

(√4+i)(√4-i)=4-√4 i+√4 i+1=5

Factorization of 6

(√5+i)(√5-i)=5-√5 i+√5 i+1=6

Factorization of 7

(√6+i)(√6-i)=6-√6 i+√6 i+1=7

[YesNo]

In the real numbers or the rationals or the complex numbers, which are all fields, you can factor 5 in many ways. Everything in those sets (except 0) has a multiplicative inverse. They are all units. There are no primes or irreducibles.

It is only when you have a set like the integers or the Gaussian integers or even some set of algebraic integers such as the integers with sqrt(-5) added to them, that you don't have multiplicative inverses for everything. Because not everything (except 0) has a multiplicative inverse there are elements that could be called irreducible, or if the ring is a unique factorization domain, a prime.

[desiresjab]

I tried factoring the number 6 down further because:

The factorization of 2=

(√1+i)(√1-i)=1-(√1-i + √1-i)+-(i)2=1+1=2

And the factorization of 3=

(√2+i)(√2-i)=2-(√2 i+√2 i)+1=3. Therefore

(√1+i)(√1-i) x (√2+i)(√2-i),

is just another expression of 2 times 3, and does work by my clumsy calculations, but only 8 of the possible 24 permutations of the factors, which we can name ABCD, will produce 6.

(AB)CD, (AB)DC, (BA)CD, (BA)DC and the palindrome (here I mean reverse order) of each works, for a total of eight. Any other order of factors does not produce 6 for me, but those eight orders do. I do not know if this is enough to be defined as a generalization of commutivity in the Complexes, or if it signifies the Complex version of non-commutivity. From an integer standpoint it seems some commutivity and some non-commutivity are involved.

[desiresjab]

In the real numbers or the rationals or the complex numbers, which are all fields, you can factor 5 in many ways. Everything in those sets (except 0) has a multiplicative inverse. They are all units. There are no primes or irreducibles.

It is only when you have a set like the integers or the Gaussian integers or even some set of algebraic integers such as the integers with sqrt(-5) added to them, that you don't have multiplicative inverses for everything. Because not everything (except 0) has a multiplicative inverse there are elements that could be called irreducible, or if the ring is a unique factorization domain, a prime.

I may require a slight perspective change. Water is trying to soak into plastic here. It just may be that I am so used to thinking of commutative rings with a modulus involved, that I need to step back and temporarily release the notion of a modulus ruling the ring, to see ideals more clearly. With a prime modulus in normal integers every element of the ring will have an inverse and no two elements the same inverse, I believe, as well. In a modulus ring of integers there is some notion of divisibility, whereas if one must strictly be in Integers, no inverse could exist. But stick a modulus anywhere and some inverses will appear.

[YesNo]

If one is thinking of a modulus one might be in a field of equivalence classes of integers rather than the integers themselves. For example, Z5 would contain five equivalence classes or sets of integers. Each would have a different remainder in the integers modulo 5. This would be a field since every equivalence class or element of that field except the 0 equivalence class would have an inverse. There would be no primes or irreducibles in that structure.

It might be interesting listing different fields and then different kinds of rings that are not fields just to get a set of examples to work with.

The fields would contain the following: (1) real numbers, (2) rational numbers, (3) complex numbers, (4) equivalence classes of integers modulo a prime integer. The only ideals here are the zero ideal (0 element) and the unit ideal (entire field).

The rings would contain the following: (1) integers, (2) Gaussian integers, a + bi where a and b are integers, (3) rings of various algebraic integers, like the Gaussian integers but instead of i = sqrt(-1) some other root of a monic polynomial, (4) polynomials with rational coefficients. Rings should provide interesting examples of ideals.

There are probably many others. The above are all commutative and so there must be some non-commutative examples such as matrices.

[desiresjab]

Being a ring or a field depends on the number of defined operations, I believe. Fields, as I take it, usually have four defined operations, rings two or three operations.

[desiresjab]

You said earlier that all interesting examples of ideals were non commutative. I do not see why that is true. I thought there were interesting examples in both, but my vision will develop as I continue. I do need to shed the habit of always thinking in modulus rings.

My thinking was temporarily derailed over the last few days by music. I made a few posts in a music thread and that got me listening again. I have to stay away from music or it derails everything else and takes over my intellectual life. I unfortunately have to segregate all of my interests like that. I can only do one field at a time.

I cannot live on human diurnal schedules. My cycle is about 41 hours instead of 24. My overlap is congruent to 17 (mod 24), with irregularities associated with obligations. Night and day are arbitrary to me. I got that way from decades of marathon poker sessions.

[desiresjab]

(√4+i)(√4-i)=4-√4 i+√4 i+1=5, also 1(5)=5

(√6+i)(√6-i)=6-√6 i+√6 i+1=7, also 1(7)=7

* * * * *

You assert there are more (in fact infinite) factorizations of 5. What are some? At the level above, the difference between 4n+1 and 4n+3 numbers is not obvious. The "splitting," they do must be visible from another vantage point.

[YesNo]

I don't know if the interesting examples of ideals are in commutative or non-commutative rings. At the time I was only able to think of commutative examples.

I think it would depend on the ring or field whether there were infinitely many different elements that could divide into 5. In a finite field there would only be finitely many different elements to divide into 5. In the rationals, one could take every rational number and divide it into 5. These aren't really interesting because in the field 5 is a unit. It is not a prime.

One could say that (1)(5) = 5 is a factorization, but the 1 is a unit. If one restricts the factors of 5 to be irreducibles or primes, then some rings such as the Gaussian integers would be able to factor 5 and other rings such as the regular integers would not.

Edit: Here's a list of algebraic structures. I have not studied most of them, but it is good to see them in one place with links: https://en.wikipedia.org/wiki/Algebraic_structure Maybe one day, I'll look at them more closely.

[desiresjab]

One thing we can be sure of: the concept of left and right ideals is applicable only for non commutitive rings. In a commutative ring such as a modulus ring, left and right ideals are the same, since x times r and r times x are not different. So any time they start tlking about left and right ideals or splitting, you know they are talking about some non commutitive object.

Everywhere I turn are statements I do not understand. All I can do is take them one at a time, putting them on the hold list until I can get to them. For instance, why and how there are exactly 21 different quadratic fields is still quite a mystery to me.

[desiresjab]

Since ideals at heart are instances of multiples, we always need to be able to see how any ideal is a multiple. Any time that becomes clear, we have understood the ideal. Even with Carmichael numbers the quest can be reduced to finding and understanding what this multiple is of. What is the generator and what does it leave in its tracks?

In the case of Complex numbers there have to be two generators, as I see it, one for the integral x-axis and one for the imaginary y-axis. Even if they do not show it in many diagrams, the y-axis is really the i-axis, 1i, 2i, 3i etc. An example would be the lattice diagram in the following link:

http://mathworld.wolfram.com/Ideal.html

[desiresjab]

In the case of Complex numbers there have to be two generators, as I see it, one for the integral x-axis and one for the imaginary y-axis...http://mathworld.wolfram.com/Ideal.html

I am not too sure about my statement here. I mean, I still think there are two generators (are they called units, or not?), but I am not sure they are assigned to each axis. (1+i) is not even on an axis.

* * * * *

My oversight recently that not every Gaussian integer is a root of an equation, or something like that, was to due to a mix up with something I read. Even as I wrote it I knew it must be wrong, but I wrote it anyway because I thought I had read it on a long-time-without-sleep binge just prior to that. I must apologize for that embarrassment. I don't really know what I confused it with either.

* * * * *

On to the more interesting question of ways to factor 5 in the Complex numbers.

We already know √4+i)(√4-i)=4-√4 i+√4 i+1=5.

To write (2+i)(2-i)=4-2 i+2i i+1=5, is essentially a trivial change from √4 to 2, so is cheating and not a valid different factorization. But how about this?:

(1+2i)(1-2i) = 1+-2i+2i-4(i)2 = 1+4 = 5

That is definitely a different factorization of 5. Are there others? Maybe. I have not validated your claim yet. All it takes is the above to show lack of unique factorization. I am simply curious if there are more, or infinitely many, as you said, I believe.

Hmmmm... I think it may be the case that these factorizations display a type of symmetry that is important later on, where exchanging a and b in the Gaussian integer does not change the result of an equation containing them. What we did in the factorizations above is exchange a and b.

But somehow I feel I can make it work for 7 as well, which is supposed to be a Gaussian prime. Let's take a look.

7 = √6+i)(√6-i)=6-√6 i+√6 i+1=7

Doesn't exchanging a and b have to work?

(1+√6i)(1-√6i)=1-√6i+√6i+6=7. Yes, it works.

Now I really am confused, I thought 7 was a Gaussian prime, but I have found two different ways of factoring it that seem distinct. The method should work on any Gaussian integer, in fact. These two factorizations do not seem trivially different. What is going on???????

[desiresjab]

Here is another factorization for 7.

(2+√3i)(2-√3i)=4+3=7.

Maybe those factors are not irreducible, so this factorization would not count. I do not know for sure. But there it is anyway, another factorization of 7. I could not find another one for 5.

[YesNo]

On to the more interesting question of ways to factor 5 in the Complex numbers.

We already know √4+i)(√4-i)=4-√4 i+√4 i+1=5.

There are infinitely many ways to factor 5 in the complex numbers. Let c be a complex number. Since the complex numbers are a field, 5/c = d is a complex number. Multiply both sides by c and get 5 = cd.

However, if one restricts attention to only the Gaussian integers https://en.wikipedia.org/wiki/Gaussian_integer, that is complex numbers like a + bi where a and b are in Z, then 5 should have a unique factorization into irreducibles (primes).



To write (2+i)(2-i)=4-2 i+2i i+1=5, is essentially a trivial change from √4 to 2, so is cheating and not a valid different factorization. But how about this?:

(1+2i)(1-2i) = 1+-2i+2i-4(i)2 = 1+4 = 5

That is definitely a different factorization of 5. Are there others? Maybe. I have not validated your claim yet. All it takes is the above to show lack of unique factorization. I am simply curious if there are more, or infinitely many, as you said, I believe.

However, you seem to have two different factorizations above. I will have to check this further. Also, I am not sure why the Gaussian integers are a principal ideal domain which would make it be a unique factorization domain. So I will look up some proof for that as well.



Hmmmm... I think it may be the case that these factorizations display a type of symmetry that is important later on, where exchanging a and b in the Gaussian integer does not change the result of an equation containing them. What we did in the factorizations above is exchange a and b.

But somehow I feel I can make it work for 7 as well, which is supposed to be a Gaussian prime. Let's take a look.

7 = √6+i)(√6-i)=6-√6 i+√6 i+1=7

Doesn't exchanging a and b have to work?

(1+√6i)(1-√6i)=1-√6i+√6i+6=7. Yes, it works.

Now I really am confused, I thought 7 was a Gaussian prime, but I have found two different ways of factoring it that seem distinct. The method should work on any Gaussian integer, in fact. These two factorizations do not seem trivially different. What is going on???????

In the case of 7, note that √6 is not an integer, that is, an element of Z. Therefore √6+i and √6-i are not Gaussian integers, but complex numbers. In the field of complex numbers 7 is a unit and everything divides it, but not in the ring of Gaussian integers.

[desiresjab]

There are infinitely many ways to factor 5 in the complex numbers. Let c be a complex number. Since the complex numbers are a field, 5/c = d is a complex number. Multiply both sides by c and get 5 = cd.

However, if one restricts attention to only the Gaussian integers https://en.wikipedia.org/wiki/Gaussian_integer, that is complex numbers like a + bi where a and b are in Z, then 5 should have a unique factorization into irreducibles (primes).



However, you seem to have two different factorizations above. I will have to check this further. Also, I am not sure why the Gaussian integers are a principal ideal domain which would make it be a unique factorization domain. So I will look up some proof for that as well.



In the case of 7, note that √6 is not an integer, that is, an element of Z. Therefore √6+i and √6-i are not Gaussian integers, but complex numbers. In the field of complex numbers 7 is a unit and everything divides it, but not in the ring of Gaussian integers.

Oh, yes, that is correct, they are complex numbers, not Gaussian integers.

[desiresjab]

On your next to last comment in your last post, I comment: All principal ideals are bsed on the multiple concept, if I have my reading straight this time. I have a sneaking suspicion that there are ideals based on other properties than simply "being a multiple of." I have a hunch Carmichael numbers might be non-principal ideals. But I seem to be about 50-50 on the hunches these days.

[YesNo]

An ideal is also an additive subgroup. From that perspective, if there is more than one generator then one has to also consider not only the multiples of each of the generators, but also sums of anything generated from those two or more generators. One could generate the even numbers in the integers Z by using the principle ideal generated by 2 or the non-principal ideal generated by (2,4). However, those ideals are the same.

Your earlier observation about factoring 5 in two different ways still has me puzzled.

Edit: I think this resolves my earlier puzzlement:

The two factorizations of 5 provided earlier are the same up to units in the Gaussian integers. So unique factorization, up to units, still holds.

To see the significance of this, look at 10 in the integers. This factors as 10 = 2*5 but also as 10 = (-2)*(-5). Those are two different factorizations but not up to units since if I multiply (-2)*(-5) by 1 = (-1)*(-1) then (-2)*(-5)=2*5.

In the case of Gaussian integers there are four units: 1, -1, i, -i. If I multiply 1+2i by 1 = (-i)(i), I get (-i)(i)(1+2i) = i(-i+2) = i(2-i). If I multiply 1-2i by 1 = (-i)(i), I get (-i)(i)(1-2i) = -i(i+2) = -1(2+i). So with 1 = (i)(-i), I get 5 = (2+i)(2-i) = 1(2+i)(2-i) = (i)(-i)(2+i)(2-i) = i(1-2i)(2-i) = i(2-i)(1-2i) = (1+2i)(1-2i).

So the two factorizations are the same up to units.

[desiresjab]

All right, then.

* * * * *

(√5+i)(√5-i)=5-√5 i+√5 i+1=6

(√1+i)(√1-i)(√2+i)(√2-i)=6

We know the second factorization is not fully commutable.

* * * * *

These factorizations are in complex numbers. We can be in complex numbers, it is a number system. Are we ever in Gaussian integers in that sense? I just see them as a complex number with the form xn-1, not as a number system. Are we ever really in Gaussian integers, other than to examine one or a few? They are just a subset of the complex numbers, are they not, perhaps part of some quadratic field?

[YesNo]

If we restrict ourselves to using only elements from the integers then we are in the integers and can see what algebraic structure exists there (commutative ring). The same thing goes for the Gaussian integers. Just use only Gaussian integers. In the integers or the Gaussian integers one can talk about unique factorization and primes.

Now if we expand the set to include all the multiplicative inverses, we move from integers to rationals (or reals) or from Gaussian integers to complex numbers, then one loses the idea of prime and unique factorization, but one gets all those multiplicative inverses and the algebraic structure is different (field).

Alternatively one can think of elements in the set of Gaussian integers as not being complex numbers. This would justify having a different name for them even though they look the same and have many other properties of addition and multiplication in common.

[desiresjab]

Aren't Gaussian integers only a tiny slice (an infinite tiny slice) of the complex field? Only a few numbers are of the the form xn-1. Or is that xn+1? Christ!!!

Before, I was kind of thinking of Gaussian integers as just another name for Complex integers. I can see that is dead wrong. There are relatively so few Gaussian integers (in the sense of density) that I don't see what good they are. Anything recovered there would be for only a few numbers. In fact, the Gaussian integers seem really, really, really restricted.

I guess I will have to get used to tooling around in these various subsets and nail down their identities better. It turns out I do not yet know exactly where I am.

[desiresjab]

Wait! Wait! Wait! Wait! Wait! Where do I get my stupid notions from sometimes? I think from laziness, which looks at things too casually and without enough effort.

A Gaussian integer is simply a Complex number the coefficients of whose real and imaginary parts are both rational, in fact both integers. They are not that sparse, then. They will fall right where integers fall on the Cartesian plane. And didn't Gauss himself say that a complex number whose both parts are rational is an integer? He manages to get much out of nothing. These are merely the complex numbers that are just the regular integers, aren't they, the ones able to shed the extra clothing of a complex number and look like a normal integer--well, behave like one, too?

I am seeing it much better now, actually. I seem to need rescuing from misconstruction more often these days as the iron gets heavier.

[YesNo]

Yes, the Gaussian integers are those complex numbers where both the real and imaginary parts are integers, such as, 3 + 5i or 250 - 26345i.

I am studying Game Theory because my daughter is taking a class in it and she discusses it. Also, I've picked up a couple of books from the library on fractals. I want to see to what extent a market chart can be represented as a self-affine fractal and where that breaks down. The fractal should be random but the market chart is not according to socionomics although fractals can be used to model them.

[desiresjab]

Yes, the Gaussian integers are those complex numbers where both the real and imaginary parts are integers, such as, 3 + 5i or 250 - 26345i.

I am studying Game Theory because my daughter is taking a class in it and she discusses it. Also, I've picked up a couple of books from the library on fractals. I want to see to what extent a market chart can be represented as a self-affine fractal and where that breaks down. The fractal should be random but the market chart is not according to socionomics although fractals can be used to model them.

I love how complex functions act, I just do not like calculating them very much. It really is a job for computers, being more messy on every level. The Euclidian algorithm is more messy, too, and correspondingly easier to make a mistake in. I am not tempted to do many of these long-winded (ugly ) calculations. They are, in fact, almost precisely what I love to stay away from in math when I can and still understand. However, it is desirable to be able to distinguish Gaussian primes quickly from other numbers that might resemble one but are not. It could easily become necessary to get down in the mud of Gaussian integers and do the arithmetic in some context or other.

* * * * *

I have read a little on both Game Theory and Fractals. Can't help you, really. I do know stock prices were there right from the beginnings of the theory, however, as they were involved in some observations of a few pioneers, Lorentz among them, I believe. Now I remember, he happened to see a chart of historical cotton prices and noticed how similar it was to long term weather patterns.

[YesNo]

I think the Gaussian primes would be either primes from the integers with a remainder of 3 mod 4 or the factors in Gaussian integers of primes from the integers with a remainder of 1 mod 4. However, I don't know if that is true or not. That is, I can't think of how I would try to prove that at the moment, in particular prove that those two characteristics give all the Gaussian primes.

[desiresjab]

The 4n+3 primes come as they are. 4n+1 primes from the integers can be broken back to smaller prime factors. Even 2 is not prime in that world because it can be factored as (1+i)(1-i).

Does that mean there is no one-to-one correspondence between primes in the integers and primes in the Gaussian integers? 5 is made of two factors, for instance. The density of Gaussian primes must be slightly higher because of that.

[YesNo]

I think one could put the Gaussian primes into a one-to-one correspondence with the integers although I don't have a function that would do that at the moment. But that just counts them. There may be some other "measures" associated with them outside of quantity.

Here is some suggestion how to proceed constructing a one-to-one correspondence. First use the one-to-one correspondence between the integers and the rational numbers. Then map the Gaussian integer (or just prime), represented as a+bi, to the rational number represented as a/b. Those two one-to-one correspondences may be able to be composed in some way to get the desired one-to-one correspondence to show the primes and the Gaussian primes have the same countable infinity of elements.

[desiresjab]

If the normal primes have density 1, the Gaussian primes should have density 2.

[YesNo]

What is "density"? One might be able to look at that as the number of primes in some region. It reminds be of big-O estimates of the number of primes less than a certain number n.

[desiresjab]

Tired of coming on here to be told for no reason I am yet again barred.

[desiresjab]

There had ought to be 3/2 as many primes in the Gaussians as there are in the integers, since every 4n+1 prime in the integers has two prime factors in the Gaussians, is what I meant by density.

[YesNo]

I think one could define something like that density if one asked how many primes are less than the norm (or absolute value in integers). There might be more than 3/2 primes in the Gaussian integers less than a certain norm n since they are contained in a plane rather than just a line, but maybe not.

[desiresjab]

I just meant to give a rough idea that there are more, or at least seem to be.

* * * * *

Well, it is very interesting that ideals somehow manage to recover unique factorization (though only in some limited cases, according to Wildberger, he called them quadratic instances, or something like that), so now the aim has to be to produce a specific example of how ideals managed to recover uinique factorization. After that, it is on to Carmichael numbers and the understanding of how they have anything to do with ideals. That is the plan anyway and as far as I can see right now. Something always gets in the way of direct progress.

[YesNo]

I don't know how ideals do that either. That does seem like a major justification for considering them.

[desiresjab]

I don't know how ideals do that either. That does seem like a major justification for considering them.

Ideals seem to eliminate the difference between 4n+1 primes and 4n+3 primes. That is, every ideal generated by a prime is a prime ideal regardless of its type to begin with. This is really cool, even if it only provides some unique factorization domains. The fact that any part of unique factorization has been recovered has got to be one of the greatest acheivements of mathematics.

[YesNo]

The Gaussian integers are a unique factorization domain already. So, the 4n+1 or 4n+3 primes are not a problem. We just get different primes than we might have expected in the Gaussian integers.

But the integers with the sqrt(-5) are not a unique factorization domain. So here is where the ideals should help, but I don't see how at the moment.

[desiresjab]

The Gaussian integers are a unique factorization domain already. So, the 4n+1 or 4n+3 primes are not a problem. We just get different primes than we might have expected in the Gaussian integers.

But the integers with the sqrt(-5) are not a unique factorization domain. So here is where the ideals should help, but I don't see how at the moment.

In red, are you talking about something like 2+i and 2-i as primes?

There is a bit of mystery here. I am wondering why if the Gaussians are a UFD is there more than one way to factor numbers such as 6 or 5 within it? I can almost trust I am overlooking something.

Anyway, key things get remembered. Another one is

Quotinet rings divided through by a maximal ideal produce a field.

Since the field seems to consist of only 0 and 1, I can't see yet why that is so important, but it seems to be.

Here is another key to hold onto:

Ideals are to rings as normal subgroups are to groups.

That shouts: Go study groups, doesn't it?

I hope that is correct. There is a lot of talk of cosets, too, and I think that may be more group theory in spite of the name recalling set theory.

* * * * *

My experience with being slow tells me a clear understanding of ideals is in front of my face, unrecognized, while I acclimate my brain to something new. All these little brealkthroughs will eventually amount to sort of an ephiphany.

The idea of principal ideals is pretty clear--they are just multiples. Like rings, ideals need 0 and 1 in the set--maybe some unit that stands in for 1, since the set of even numbers, for instance, cannot have a 1 in it.

Ideals are presented in additive terminology, though they have multiplicative properties.

I think this idea of "splitting," I keep running into refers to non-commutivity. I think this splitting is related to factorization problems. My present guess is that ideals manage to bypass this problem. I think this problem is related to the fact that 4n+3 primes are also primes in the Gaussians but 4n+1 primes are not, as they can be factored into smaller factors.

I hope I am not too amiss here. I am trying to put my collage together.

[YesNo]

In red, are you talking about something like 2+i and 2-i as primes?

There is a bit of mystery here. I am wondering why if the Gaussians are a UFD is there more than one way to factor numbers such as 6 or 5 within it? I can almost trust I am overlooking something.

Up to units, there is only one way to factor 6 or 5 in the Gaussian integers. For example one can factor 6 = (3)(2) in Z. But one can also factor it as (-3)(-2). One could also reorder the factors as (2)(3). These are all different factorizations, but they are not what unique factorization tries to capture as an idea. The unique factors do not depend upon the order of the factors, nor do they depend on whether one can multiply the factors by 1 and get a different set of factors (associates). Again using 6 in Z, we can write 6 = (2)(3) = (1)(2)(3) = (-1)(-1)(2)(3) = (-2)(-3). In the Gaussian integers there are four units, not two as in Z: 1, -1, i, -1. Their norms are all 1. That challenges our normal intuition about what a unit should be (not just 1 or -1) and what an associate factor would be (not just multiplying the factor by -1).



Anyway, key things get remembered. Another one is

Quotinet rings divided through by a maximal ideal produce a field.

Since the field seems to consist of only 0 and 1, I can't see yet why that is so important, but it seems to be.

Here is another key to hold onto:

Ideals are to rings as normal subgroups are to groups.

That shouts: Go study groups, doesn't it?

I hope that is correct. There is a lot of talk of cosets, too, and I think that may be more group theory in spite of the name recalling set theory.

* * * * *

My experience with being slow tells me a clear understanding of ideals is in front of my face, unrecognized, while I acclimate my brain to something new. All these little brealkthroughs will eventually amount to sort of an ephiphany.

The idea of principal ideals is pretty clear--they are just multiples. Like rings, ideals need 0 and 1 in the set--maybe some unit that stands in for 1, since the set of even numbers, for instance, cannot have a 1 in it.

Ideals are presented in additive terminology, though they have multiplicative properties.

I think this idea of "splitting," I keep running into refers to non-commutivity. I think this splitting is related to factorization problems. My present guess is that ideals manage to bypass this problem. I think this problem is related to the fact that 4n+3 primes are also primes in the Gaussians but 4n+1 primes are not, as they can be factored into smaller factors.

I hope I am not too amiss here. I am trying to put my collage together.

I am not clear about all of this either and I am finding it interesting to get a better understanding. In the Wikipedia article, https://en.wikipedia.org/wiki/Unique_factorization_domain , there is a class chain:


commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields


There should be examples of non-unique factorization in structures to the left of "unique factorization domains" but none to the right. There should also be examples of a unique factorization domain that is not a principal ideal domain which is where the prime ideal questions we are discussing seem to be most important. But I am still unclear about how to formulate the questions.

[desiresjab]

Speaking of prime factorization in the Gaussians, you must be right. But in the Complex numbers is 2 even a prime? I can factor it as

(1+i)(1-i).

5 =(2+i)(2-i). Those are Gauusian integers, are they not (for a and b are both integers)? I believe those factors are not units, either. Does this make 5 not a prime in Gaussian integers. I believe no 4n+1 prime is a prime in the Gaussian integers, but I could be confusing Gaussians with the Complex numbers in general.

Now for 6 I really do not understand why 2x3 would be a prime factorization in the Gaussian integers, since I do not even believe 2 is a prime in that set. Isn't this the prime factorization?

(3)(1+i)(1-i)=6

What am I not getting about units for asking this question?

[desiresjab]

I have been looking around for an example of a non principal ideal. Of course it has been sitting in front of my nose.

All x+1 seem to form a non principal ideal. A non principal ideal is the kind I suspect Carmichael ideals to be.

[YesNo]

Speaking of prime factorization in the Gaussians, you must be right. But in the Complex numbers is 2 even a prime? I can factor it as

(1+i)(1-i).

In the complex numbers there aren't any primes. All are units since every complex number has a multiplicative inverse. So 2 is not a prime in the complex numbers. It is a prime in the regular integers denoted by Z, but it is not in the Gaussian integers because of the factorization you mentioned above. In the Gaussian integers, unlike the complex numbers, 2 is not a unit because 1/2 = 1/2 + 0i does not exist in the Gaussian integers since 1/2 is not an integer in Z.



5 =(2+i)(2-i). Those are Gauusian integers, are they not (for a and b are both integers)? I believe those factors are not units, either. Does this make 5 not a prime in Gaussian integers. I believe no 4n+1 prime is a prime in the Gaussian integers, but I could be confusing Gaussians with the Complex numbers in general.

In the Gaussian integers you have factored 5 into two other Gaussian integers. So 5 is not a prime in the Gaussian integers. In the complex numbers 5 is not prime either, but that is because it is a unit. It has a multiplicative inverse 1/5 + 0i in the complex numbers, but that inverse is not in the Gaussian integers because both a and b in a + bi have to be normal integers that one has in Z.



Now for 6 I really do not understand why 2x3 would be a prime factorization in the Gaussian integers, since I do not even believe 2 is a prime in that set. Isn't this the prime factorization?

(3)(1+i)(1-i)=6

What am I not getting about units for asking this question?

Right, in the Gaussian integers 6 factors as you mentioned. In the normal integers or Z, 6 = (2)(3). I was using that factorization to show what an associate was. In Z 2 has an associate -2. In the complex numbers 2 has -2, -2i and 2i as associates, since I just multiplied 2 by all the units in the Gaussian integers (1, -1, i, -i).

[YesNo]

I have been looking around for an example of a non principal ideal. Of course it has been sitting in front of my nose.

All x+1 seem to form a non principal ideal. A non principal ideal is the kind I suspect Carmichael ideals to be.

A non-principal ideal for this polynomial ring over the integers would be (x, 2), that is, the ideal generated by x and 2. This is not the whole polynomial ring. That would be generated by a unit, say 1, but it contains those polynomials where the constant term must be even. It cannot be reduced to a principal ideal because then we would need some polynomial that divided both x and 2 (besides a unit like 1) so we could get x and 2 as a product. If such a thing existed it would be both an integer > 1 and a polynomial of first degree in x which is a contradiction. So this can't be reduced.

This link might be useful in terms of making sense how ideals resolve the problem of not having unique factorization. These are just notes used with a larger text that popped up from an internet seach and I don't know who the author is, but the first few pages seemed to summarize the problem: http://www2.math.ou.edu/~kmartin/nti/chap11.pdf

[desiresjab]

That link was helpful in that it knocked a few chips off a mountain of marble and raised more questions than it answered. I must say the notation and terminology is (to resort to Trumpian irreducibles) very bad; very, very bad, and the people who devised it were some of the worst people.

That is still my gripe about mathematics--the same notation and terminology mean different things in different areas of math, sometimes closely related areas! The job of revision is too big for me, but I may get stubborn and refer to ideals like this <1+√5> in the future, instead of with parentheses which are so easily confused with standard multiplication. Later on, the author of that paper admits the terminology and notation in ideal theory has evolved in an unfortunate manner.

I wonder how many graduate students have sat in upper level math classes praying for a concrete example using almost all numbers and a minimum number of letters in place of numbers, just so they can establish through a concrete example what the hell is being generalized, and how so, in the first place?

* * * * *

Some of the pieces I am chipping off are above my current understanding and stored away until such time as they fit into a total picture. The idea of division in ideals is completely queer to me as I view it presently. It doesn't even seem like division. Division in ideals seems like the reciprocal of division as I know it but not multiplication.

What I want to do is go back to a more basic level.

* * * * *

Back to 2Z, the good old even integers. Is this the set that forms the ideal of 2Z?

{0, 1, 2, 4, 6, 8,...}.

I am not sure if I have to notate it somehow as 0 and 1 adjoined to this infinite set of even numbers or not. From everything I have read I am led to think that

{0, 1, 2, 4, 6, 8,...}

would be the complete set of ideals for 2Z, 0 being the 0 ideal (additive identity), 1 being the unit (multiplicative identity), and 2 being the generator or principal ideal of the set. Each of the elements is called an ideal, 2 being the principal ideal.

So what is the entire set itself called? Is it called the set of ideals? If so, then I suppose the set of ideals of 2Z, or something like that? I am a bit confused on these several points, so I hope you can answer them.

Once I have these issues straight, I can move on to my next issues.

[YesNo]

I have to go to a Chinese New Year celebration, so I'll get back to you either this evening or tomorrow.

I agree that examples are always good. The simpler the better.

The ideal (2) in the integers is the set of all even numbers, positive, negative and 0 with the operations that one has in the integers. Are you referring to the finite field containing two elements 0 and 1 which can be viewed as equivalence classes of even and odd integers?

Since an ideal is a set of elements from the ring with ring operations, division of these objects would be viewed as set inclusion.

[desiresjab]

We could speak in general of the ideals generated by the ring of integers Z, where Z is really 1Z where the 1 has been dropped. The numbers generated when 1 is used as the generator are just 1∙1, 1∙2, 1∙3, 1∙4,..., in other words just the full set of integers, which contains within itself all subsets of multiples of every integer, since those multiples are just integers, too.

The ideal 2Z is the set of all even integers, plus 1 and 0, because an ideal always has to have 1 and 0 in its set.

We could have 4Z, where the ideal would be all multiples of 4 from the integers, plus the elements 1 and 0 again, adjoined or however it might be expressed in the case of 0.

An ideal is not maximal if all the elements of the ideal are found in a larger subset of Z which is smaller than Z itself. In the case of 4Z, all integers which are multiples of 4 would be contained in the larger set of merely even numbers 2Z, which lies strictly between the multiples of 4 and the multiples of 1. Therefore 4Z is not a maximal ideal, because all its elements can be found in 2Z. However, 2Z is a maximal ideal because no subset smaller than Z itself "contains" it. This use of contains actually means divides in the language of ideals, according to the link recently posted by Yes/No.

It turns out that maximal ideals are generated by ordinary prime numbers, and only by them, no distinction being made between 4n+1 primes and 4n+3 primes.

I believe all maximal ideals are prime ideals, but 0 is a prime ideal, too, so all prime ideals are not maximal.

[desiresjab]

The ring of integers (mod p) where p is a prime is actually a field, a finite field. These fields are cyclic. I am not sure what precisely makes it a field instead of a ring, since I have been limping along under the impression that all four arithmetic operations had to be fully defined for fields. Well now, it occurs to me based on what Yes/No recently said about inverses, that as long as the modulus is prime every element of the residue system will have an inverse. Since every element has an inverse, that must be what makes it a field. They call that every element being a unit, which is a new one on me.

Under a modulus that is prime where every element has an inverse, I suppose that is enough to fully define division, which also would make finite fields under a modulus fields instead of rings. Somehow, I think they are fields and rings at the same time, if the definitions are not strictly mutually exclusive, which they could very well be.

I have been studying all day. I do not want to look this latest item up. Yes/No will know the answer. I will let him answer it. I want to watch movies right now.

[desiresjab]

Ah, yes, finally many things are coming together at once. Maximal ideals are indeed always prime ideals.

And, no, it is not impossible to be a Field and a Ring at the same time. A spoken proof sees that all Fields are Integral Domains, but the definition of Integral Domain explains it as a particular type of Ring.

The integers (mod p) where p is a prime, which we are so familiar with, forms a commutative ring where no two elements (residue classes) multipled together equals zero. The cyclic elements also form a Finite Field, and finite or not, all fields are integral domains, which are also rings.

In the language of ideals the integers mod (n) are called a Quotient Ring. This is what we are most familiar with in different language. Still, it is best not to lean too heavily on this particular understanding, for there is nothing cyclic about ideals in general. It does, however, apply because we would not be amiss in thinking of a quotient ring simply as the integers (mod n), where n is not necessarily a prime, as long as we do not forget that an object such as a Quotient Ring can be and is constructed not only from integers but from polyniomials as well, so that one has to remain aware of which type an article or portion of an article is alluding to.

* * * * *

Rings of polynomials are more difficult to deal with than integers algebraically. The dudes who developed the theory were high-powered intellects at the kind of old style algebraic stuff that mathematicians like Euler and Jacobi were known for--those long painful derivations and such that one amazes anyone ever unpuzzled for the first time. We have not gotten our hands dirty yet with manipulating polynomials. We might have to do that some time, or perhaps it can be avoided.

I hear other areas calling, so I want to get this one settled up as soon as I can. Combinatorics I already have pretty good basic experience and knowledge of in the sense that I played poker for many years and I was the kind who liked to calculate a lot of things out. Fancy counting is a load of fun, as I see it. I want to do more and see how it relates back to some of the stuff in number theory we have been looking at. We know for a fact that every group is a permutation group. I think Cayley proved that.

[desiresjab]

Yeah. Combinatorics, there is a lot more I want to learn there. In complex combinatorial situations, getting the logic just right is hairy. I love that stuff. Right now I am stuck here with ideals. But that is good. I only get stuck where I have chosen to get stuck already, for any one place that one settles in to confront higher mathematics there will be challenges that block progress for a while. Though I have in the last day made many connections that have avoided me, still I have a long way to go with ideals before I will be satisfied. The final capper will have to be an exposition of Carmichael ideals.

[desiresjab]

We can relax a bit now and give an entertaining question I remember from somewhere in the past that has to do with probability and game theory.

Three men with rifles are arranged in an equilateral triangle, all an equal distance apart. They are going to fire at each other in turn, until only one remains.

Participant A has a 90% accuracy rate, and participant B has a 70% rate. Poor C is a lousy shot, hits his target only 30% of the time, and is first to fire. What is his best strategy?

[YesNo]

We could speak in general of the ideals generated by the ring of integers Z, where Z is really 1Z where the 1 has been dropped. The numbers generated when 1 is used as the generator are just 1∙1, 1∙2, 1∙3, 1∙4,..., in other words just the full set of integers, which contains within itself all subsets of multiples of every integer, since those multiples are just integers, too.

The ideal 2Z is the set of all even integers, plus 1 and 0, because an ideal always has to have 1 and 0 in its set.

We could have 4Z, where the ideal would be all multiples of 4 from the integers, plus the elements 1 and 0 again, adjoined or however it might be expressed in the case of 0.

An ideal is not maximal if all the elements of the ideal are found in a larger subset of Z which is smaller than Z itself. In the case of 4Z, all integers which are multiples of 4 would be contained in the larger set of merely even numbers 2Z, which lies strictly between the multiples of 4 and the multiples of 1. Therefore 4Z is not a maximal ideal, because all its elements can be found in 2Z. However, 2Z is a maximal ideal because no subset smaller than Z itself "contains" it. This use of contains actually means divides in the language of ideals, according to the link recently posted by Yes/No.

It turns out that maximal ideals are generated by ordinary prime numbers, and only by them, no distinction being made between 4n+1 primes and 4n+3 primes.

I believe all maximal ideals are prime ideals, but 0 is a prime ideal, too, so all prime ideals are not maximal.

That is how I see it as well except for the part "an ideal always has to have 1 and 0 in its set". An ideal must have 0 in it since 0 is in the ring and so the generator times 0 will be 0. So 0 is in the ideal. However, if 1 were in the ideal, then the entire ring would be in the ideal and the ideal would be generated by 1. So, in general 1 would not be in the ideal, however, 0 must be there.

A field is a special kind of ring as you mentioned in later posts.

That's is an interesting game theory question. I don't know the answer. If I were C I would aim first at A. If I were A I would aim first at B. If I were B I would aim first at A. This would lead to a pure strategy for A, B and C for their first moves. But I wonder how to solve this in a more general way.

[desiresjab]

What you say about 1 not being in the set makes sense, however my mind believes it has read a hundred times that 1 must be there. It only has an additive identity without 1 there, no multiplicative identity. Commutative, one-sided ideals always have inverses, do they not? If 1 is not supposed to be there I need to clear it out mentally, but I need to understand how I have mistread a hundred times.

[YesNo]

I think a commutative ring must have both 0 and 1 in the ring. In Birkhoff and MacLane, "A Survey of Modern Algebra", one of the axioms of a commutative ring requires that there exists an element that is the multiplicative identity. That would be 1. The ring must also have the additive identity or 0. However an ideal need not be the full ring. For example the ideal in Z generated by 3 has all the multiples of 3. This would include 0 but not 1 which is not a multiple of 3. The ring must have 0 and 1, but not the ideal.

The ring itself is one of the ideals of the ring, which might make this confusing. That ideal which equals the ring itself must contain 1 since that ideal contains everything in the ring. But all the other ideals do not contain 1.

[desiresjab]

I think a commutative ring must have both 0 and 1 in the ring. In Birkhoff and MacLane, "A Survey of Modern Algebra", one of the axioms of a commutative ring requires that there exists an element that is the multiplicative identity. That would be 1. The ring must also have the additive identity or 0. However an ideal need not be the full ring. For example the ideal in Z generated by 3 has all the multiples of 3. This would include 0 but not 1 which is not a multiple of 3. The ring must have 0 and 1, but not the ideal.

The ring itself is one of the ideals of the ring, which might make this confusing. That ideal which equals the ring itself must contain 1 since that ideal contains everything in the ring. But all the other ideals do not contain 1.

The following has a highly interesting opening paragraph.

https://en.wikipedia.org/wiki/Subring

Many (in fact the majority) of mathematicians seem to require that a ring contain a multiplicative inverse, but there are some who dispense with the notion.

The next to last sentence in that paragraph is a killer. It says the subring may have a multiplicative identity that is different from the one for R. What the...?

Anyway, slowly this thing is making sense. I always find afterwards that I have read over the truth many times before it had enough meaning for me to be included in my picture. This theory is rife with details that mean everything. If so much information were not included in these precise abstract algebraic formulations mathematicians would not be able to handle quite complex statements with a few swipes of the chalk, as I have seen them do. The propositions at this level are literally crammed full of detailed information upon which the whole content rests. Unless you understand the details perfectly, you will get the content wrong. That is why I have to be so nit-picky right now and question everything I do not firmly understand and believe--not just until I receive the right answer from someone else, but until I understand perfectly for myself. So when I sto piss questioning of everything, you will know I have probably understood or died.

[desiresjab]

That short Wiki-peja article I linked to is full of powerful statements. Of course some of those powerful statements are not entirely clear at this point in the journey, but are quite intriguing as future references to lean on.

* * * * *

Your daughter might find interesting the answer to the question I posed about A, B and C shooting at each other in trun from an equal distance. I could not remember the exact numbers of the question, so used numbers that were safe. The general idea is there.

C shoots first with only a 30% chance of striking his target, against opponents who have respectively 90% and 70% chances of hitting their target at this distance.

C must fire into the ground. If he were to accidentally kill one of the opponents, the next shot would be fired at him from an opponent who is far more accurate. His best case scenario is that B (70% shooter) kills A (90% shooter) with the next shot. That way C gets to at least fire the first shot at his superior opponent.

[YesNo]

That makes sense that C should deliberately miss the target. I'll try to see how that game fits in with the game theory I've been reading about.

A ring without a multiplicative identity would not fit Birkhoff and MacLane's assumptions, but it shows that there are other ways to organize these algebraic structures. They mentioned that the assumption of the existence of 1 not equal to 0 was to eliminate examples of rings consisting of only the 0 element. A ring, by their definition, has to have at least two elements: 0 and 1. If the ring doesn't have an identity then the ideals would be subrings as the article mentioned whereas I would not think of ideals as rings except in the trivial case where the ideal (generated by 1) equaled the whole ring.

[desiresjab]

There seems to be some confusion even among the experts as to what is called what in the theory.

I think I am getting fairly close to a decent understanding. It really helps to work out these little inconsistencies and get everything in its proper place.

One thing that makes ideals so fun to study is that the word is not used elsewhere in mathematics with a gross of other meanings. A brand new word for a brand new idea. One could improve mathematics considerably by combing through an English dictionary and finding suitable terms to replace those that are overworked to the point of ambiguity.

[desiresjab]

Oooh! Oooh! I got something else straight that has been mystifying me, and which shows my guess was wrong about "splitting fields" being connected with 4n+1 and 4n+3 primes.

Splitting refers to factoring a polynomial all the way down to linear factors. Even if you factor it you have not split it unless all the factors are linear.

A Splitting Field involves something called Field Extensions, which are literally what they say. Mathematicians take a field like the rational numbers and "Adjoin," enough non-rational numbers to it to enable them to Split a particular ploynomial or class of polynomials into linear factors. The trick is to Adjoin just enough numbers to the original set to get the job done, instead of ending up with a much larger set which indeed gets the job of splitting done but also contains many superfluous elements. You want to adjoin the minimum number of elements to the set that enables splitting. This involves a lot of complex techniques but the basic idea is not that hard, though I imagine the idea of adjoinment in this fashion was quite revolutionary when it was first proposed. The seeds may have originated in group theory.

[YesNo]

I don't know much about splitting fields, but the idea of adding only what one needs to the base field in order to factor a polynomial over the rationals sounds interesting.

A pair of confusing terms for me is "prime" and "irreducible". I assume primes only exist in a unique factorization domain, otherwise what one gets are irreducibles which are as far as one can factor an object in the algebraic structure. At least there is a factorization even though it is not the only one.

[desiresjab]

I don't know much about splitting fields, but the idea of adding only what one needs to the base field in order to factor a polynomial over the rationals sounds interesting.

A pair of confusing terms for me is "prime" and "irreducible". I assume primes only exist in a unique factorization domain, otherwise what one gets are irreducibles which are as far as one can factor an object in the algebraic structure. At least there is a factorization even though it is not the only one.

Yes, one would think every irreducible would be a prime, but that maybe is not the case. I do not yet know enough to satisfy myself but I am running out of ideas of how to proceed. I have not seen a bit of use for primaries and semi-primes, but I suppose that lies ahead.

Once ideals are understood fully, that, I believe, is the major portion of higher arithmentic that occupied great minds in the 20th century and late 19th. Going beyond ideals may require a measure of inventiveness . A full understanding of Artin's work along with ideals would be much of what one needs. I am not shooting for Artin's work, though. He is too difficult. I would be nowhere near ready for Artin at this point.

[desiresjab]

Dr. Salomone talks about that in this video. This guy lectures at light speed, and I like that. Subjects other professors take 45 minutes to discuss he dispatches of in ten minutes. No wasted time.

According to him the only cases where irreducibility and primality do not coincide are cases that are special and "not nice." I notice such special cases are usually shoved back and relegated to a later timetable when one is supposedly more advanced. The class is here is Abstract Algebra II.

[YesNo]

I figure I won't understand anything fully, but some things will be understood enough. Eventually you should be able to read Artin's work. At the moment I don't think I would understand it either. But if we kept searching for clues, it should eventually make sense. I suspect it would take less time and effort to understand Artin than to understand Joyce's Finnegans Wake.

[desiresjab]

I figure I won't understand anything fully, but some things will be understood enough. Eventually you should be able to read Artin's work. At the moment I don't think I would understand it either. But if we kept searching for clues, it should eventually make sense. I suspect it would take less time and effort to understand Artin than to understand Joyce's Finnegans Wake.

And this is the road to Artin. It forks ahead, if anyone wants to go there. Never say never. The mathematician seeks isolated clues about the structure of the universe, the artist builds an artificial one out of the materials at hand. There is some connection. It is a long ways off for the mathematician.

The right junction could present itself. There may be a turnoff toward Brocard. For this reason squares and anything about them is good to pick up. There is so much juice of squares left in field extensions, I have to stay a while longer. I also need now to go back to that difficult paper you linked to a few weeks ago, and see if I can better understand the operation of multiplication of ideals presented there.

I am getting more secure with ideals, and the idea of a subgroup within a ring that is only reachable through the subgroup itself, which precisely corresponds to what an ideal is, I hope.

Rings are to ideals as normal subgroups are to groups. Does this mean the friendly rings referred to by Salomone are the normal subgroups of group theory, as opposed to some other kind of subgroup that is not normal, like a non-commutative one, perhaps? I am going to stop guessing, but it is an addictive habit, and seems to serve me even when I am wrong, by keeping me asking questions until I am sure about something. Right now I am not sure what I am supposed to try to be sure about next, which is less fun than knowing where you should look, which is less fun than suspecting where you should look. At the moment I neither know or suspect.

[desiresjab]

I hope my old pal Yes/No is all right. It is not like him to leave posts unanswered. I wanted to say that division of ideals is not so challenging as it seemed at first glance. They call the ideal that does the dividing larger (as in terms of a larger set), for theoretcially its elements have greater density along the number line than the ideal it is dividing, such as 2Z dividing 4Z. The first may have more "elements," per unit distrance, but we realize both sets are actually infinite so there is no real difference in their cardinality but only in their density of occurence along the number line. However, it is plain to see that 2Z has a smaller generator than 4Z of which 4Z is a multiple, so naturally 2Z can divide it evenly.

As for multiplication of ideals, I am still trying to locate a paper I recently glanced at which explains it.

If you multiply two elements from the ideal, the result ends up back in the ideal. The cool thing about ideals is that if you multiply one of its element times one of the elements in the ring that is not in the ideal, the result ends up back in the ideal, too. They call that "absorbing," multiplication. In the case of non-commutative ideals, multiplication will be absorbed from either the left or the right.

I now have to look at some details of multiplication, then actually multiply some ideals together to inspect the results, then I should be done with this part and ready to investigate the relationship of Carmichael numbers to ideals.

[desiresjab]

I myself must now be gone for a few days traveling. I will investigate ideals and multiplication before I go so I will have something to dwell on while I am away. May you all be here and in good health when I return.

[YesNo]

I think I missed your previous to last post. What you say makes sense to me. I don't know how ideals fit in with Carmichael numbers, but I heard they do in some way. I'll see if I can find out more about that.

Here's something on the multiplication of ideals that I thought was interesting:

This one shows that the product of two ideals is not their intersection, in particular, 2Z multiplied by 2Z is 4Z,but that is not the intersection of 2Z with itself: http://math.stackexchange.com/questions/151317/product-of-two-ideals-doesnt-equal-the-intersection

That question also gives a definition of what multiplying two ideals means. Let I and J be ideals and ij the product of one element from I and one from J. Consider the set of all finite sums of these kinds of pairs. That would be the product of two ideas. It has to be finite since an infinite sum would likely take one out of the ring.

[desiresjab]

That is the problem. I do not see where 2Z or 4Z is finite. Element by element multiplication makes perfect sense for a finite number of elements. How do these two ideals (any two ideals) work out be finite? I do not see that.

[YesNo]

One restricts the sums to be over a finite number of products by definition. The definition makes it finite. So we have I = 2Z = {...-4,-2,0,2,4,...}. Suppose we want to multiply the ideal I by itself. We can construct a set that contains every product, 2m * 2n, where m and n are integers. This multiplies every element in I by another element in I. Now take any finite number of those products and add them together. This becomes an element in the ideal that is formed from the product of 2Z * 2Z. The smallest positive integer in that product of ideals would be 4 and that would be the generator of the ideal. This is what we would expect because we get 2Z * 2Z = 4Z = {...-8,-4,0,4,8,...}.

Regarding Carmichael numbers the Wikipedia article says that the idea of Carmichael numbers is extended to other algebraic structures through ideals. Ideals don't help us solve problems about Carmichael numbers in the regular integers. So, ideals help generalize the idea of Carmichael numbers to other algebraic structures: https://en.wikipedia.org/wiki/Carmichael_number

[desiresjab]

Thanks again. I have to revisit an article to see if I can establish for a fact that the sum of two ideals is simply their GCD. That is very curious. Well, let me see. How about a concrete example? If I took two numbers such as 19 and 12. When I add them, their sum of 31 would actually be 1, in that case. So, yes, that seems curious. It seems modular. I can see where cycling back to 1 occurs in a cyclic ring such as a modulus, but how does that apply here? Of course the GCD of 19 and 12 separately is 1, so in that regard it makes sense. The notion here seems "not usual," and I will have to think about it some more and read some more.

[YesNo]

I think one defines the sum of two ideals as what one would get if one added one element of an ideal to an element of another ideal. So if (19) is an ideal in Z and (12) is an ideal in Z, what would be in their sum would be elements like 19a + 12b for integers a and b.

Thinking about this as a greatest common divisor, since we know that the GCD of 19 and 12 is 1, we should be able to write an equation like this: 19x + 12y = 1 for some integers x and y.

But now think of 19x as some element in (19) and 12y as some element in (12). This matches up the result of the GCD operation and the sum of two ideals.

[desiresjab]

More is becoming clear but the muddled parts still drive me nuttier. There are quite a few items in the following article which confuse me. For instance, I do not "see," why or understand statelments like

Z[√-5] is already the full ring of integers of its quotient field Q(√-5). The examples they use are always √-3 and √-5, because larger ones quickly start to become unweildly. We would like to understand the difference in behavior in this arena of 4n+1 and 4n+3 primes, for apparently there is one, is about what I could get out of page four through page five of

http://www2.math.ou.edu/~kmartin/nti/chap11.pdf

Though I always pick up additional incidental details I do not fully understand or which, on the other hand, make perfect sense to me.

[YesNo]

I don't understand this well enough at the moment. I will try to read the link again. I can see how Z[sqrt(-3)] can take advantage of ideals by calling a non-principal ideal an ideal number that doesn't exist in that set of integers and using that to get unique factorization, but that is as far as I see at the moment.

[desiresjab]

I am reporting from the site of another logjam. It surprised me to find out I did not know the proper procedure for generating the lattice on the Wolfram link:

http://mathworld.wolfram.com/Ideal.html

I can see too many ways to do it, I do not know which is the correct way. It seems to me from the theory I should be able to fill <2> and <1+i> in separately or as an active combination in <2, 1+i>. I can see how to get all the even numbers on the grid. I am not sure how to get the 1+i's beyond the inner group of 1+i, -1+i, etc. I may be more confused than I thought I was. I have cleared some path ahead but had to return to this.

[YesNo]

Since (1+i)*(1-i)=1+1=2, <1+i> should generate the same lattice as <2, 1+i>. To look at <2> separately, multiply it by all Gaussian integers in the visible part of the lattice by 2 and see where they lie. Certainly any even numbers on the real axis would be in the ideal as well as those on the complex axis. Some of the other points off the axes, but not all of them, should be present as well.

[desiresjab]

I am trying to see if I can treat <2, 1+i> like Cartesian coordinates.

Points on the a+bi axis (y-axis) are not clear to me. The red point in the upper right of the diagram looks like the Cartesian point (3, 3), but must have a different description in Gaussians. Since I do not even know what that point is I cannot figure out what to multiply by to get that point either.

[desiresjab]

Wait, I see my stupid mustake. The y-axis is not a+bi, it is the bi part alone of the expression. With that bit of foolishness out of the way, getting the points down the right way might be easier. Still don't know if I can generate them all the proper way.

[YesNo]

Right. The x axis and the y axis would be the way one plots a real function, f(x) = y. The real axis and the complex axis would be how one plots a + bi. They have different names for these axes but both share in common the need to plot something in two dimensions.

There is a chapter in Birkhoff and MacLane on algebraic number fields that I hope will resolve some confusion I am having the Z[sqrt(-5)] and Z[sqrt(-3)] and how ideals help with unique factorization.

[desiresjab]

Okay, I have shown myself how to generate every point on the lattice through multiplication. These lattice diagrams may prove to be as germane to the study of ideals as Eisenstein's lattice diagram was for quadratic reciprocity. I want to make sure I get out of it everything there is to get, for I notice I can also take the additive approach and generate the same lattice, I believe. From any even number I can step 1+i or 1-i and get the remaining points, I mean. I still do not know if one way is the preferred way to see the lattice.

[YesNo]

I don't know if there is a preferred way to see this. Understanding something at all is all I aim for when looking at something I am unfamiliar with. However, finding different ways may lead to new results. That would imply a deeper understanding.

[desiresjab]

The lousy symbol processor on this site will not let me post again. I have a long post written. I do not feel like going through it to see what this system is objecting to, when I know it is the system again, not me.

This system uses brackets heavily for special functions, so whenever you use brackets in your post, especially in conjunction with a number, the system thinks you are trying to interfere with its propietary commands or something. Tired of it. These posts are hard to write, they include a lot of thinking.

[desiresjab]

What seems queer is I do not believe I could generate points like (1+3i) and (3+i) with 2 by itself, but I can generate them all with (1+i), which is not surprising, but the fact that I can apparently generate all the evens with (1+i) as well, is, somewhat, at least. Of course, I still have to have 2 in there a bunch of times as a multiplier to accomplish this when using (1+i) as the generator.

Sorry I am having to post piecemeal what would have all been in one post.

[desiresjab]

Not sure what all this means. I feel certain key ideas have to be seen with perfect clarity, and this is one of them. Every point on this lattice has to be accounted for, using the generators given. There might be (almost certainly are) other generators that would fill in the exact same lattice, but I am only concerned with these two generators right now, what they do, and especially how they do it, whether in tandem or in isolation. See what I'm sane?

Unfortunately, what I had to leave out were the actual multiplications which showed how I arrived at each point. Frustratring.

[desiresjab]

(2-i)(1+i)=2+2i-i-i2=2+i+1=3+i

[desiresjab]

Christ, to get 1+3i, multiply 1+i and 2+2i together. What is this garbage processor?

[desiresjab]

Some legal multiplications produce results outside the lattice. Hmmm. Such as:

(1+i)(2+2i)=2+2i+2i+2i2=4i

[desiresjab]

I think I know what you are trying to see. I do not know how to see it, either, but I believe I know what you are trying to see. The extension fields of √-5 and √-3. I cannot figure out what the devil they are talking about whent they speak of the primary difference between these two. It has to do with quotient fields and every integer of their qotient field already being there, at least for √-5, or some such thing.

Mathematicians have uglied up the quotient field thing real heavy with symbols. It may just mean this:

When you use an integer as your modulus over this complex field, your remainder often comes out with an imaginary piece. Well, technically it would always be there, but invisible when its constant was 0. A quotient field is somehow connected to this idea, I believe, but I do not know how and I certainly cannot prove or demonstrate it at this point.

The objects we are studying are becoming extremely abstract. It makes you appreciate the genius of the people who got there first with only their imaginations to guide them.

[YesNo]

What seems queer is I do not believe I could generate points like (1+3i) and (3+i) with 2 by itself, but I can generate them all with (1+i), which is not surprising, but the fact that I can apparently generate all the evens with (1+i) as well, is, somewhat, at least. Of course, I still have to have 2 in there a bunch of times as a multiplier to accomplish this when using (1+i) as the generator.

Sorry I am having to post piecemeal what would have all been in one post.

You might try installing python through the anaconda distribution set. You can then create juypter notebooks and use mathjax which I think is close to LaTeX. We could share these notebooks.

The Gaussian integer 1+i should generate all the points on the lattice as you saw. Since 1+i divides 2, 2 should not generate all of the lattice points but only a subset of them. Although 4i is not on the portion of the lattice visible in the link, the lattice contains infinitely many points. It is just off the part that was shown.

[desiresjab]

Next, I sense we need to do some actual, ugly long-hand dividing in this territory to unlock some secrets and lighten some dark passage ways, but I do not even know how to start. We need the semblance of a problem to solve. Do you have a light?

[YesNo]

The way I would approach dividing is to construct the reciprocal and multiply. The reciprocal exists in the complex numbers but perhaps not in the Gaussian integers, so we can construct it.

For example suppose I wanted to divide 2 by 1+i. I would write that as 2/(1+i). But 1/(1+i) = 1(1-i)/(1+i)(1-i) = (1-i)/2. I multiplied the denominator (1+i) by its conjugate (1-i). That will give me an integer in the denominator. If I do that to the denominator, I have to do it to the numerator. That is why I multiplied 1/(1+i) by (1-i)/(1-i)=1. Now I can multiply 2 by (1-i)/2 and get 1-i.

That may not be what you are looking for. Being able to formulate a problem even if one cannot solve the problem is valuable work.

[desiresjab]

Thank you.

I think I am seeking the tie-in between addition and multiplication in ideals. I find something slightly peculiar there.

We know I can generate the Wolfram lattice with just 1+i. We know I cannot generate that lattice with merely the ideal of 2. But 2 should give me all the even points on the lattice, as I see it. From any even point I am able to reach the "odd," points simply by adding 1+i to this even value, in other words it seems just like combining the two ideals through addition, and it appears to work. I do not know if this method is valid. It seems like it would have to be. It seems like it represents that tie-in I am looking for between addition and multiplication in this relam.

Above all, ideals seem to be additive objects. I believe but cannot prove that I have now demonstrated this relationship. I have generated all the points on the lattice two different ways (separately through addition & multiplication), and I believe they are both valid, not just a coincidence. Perhaps I am wrong, but you see now what I am looking for, whereas you might not have before.

[desiresjab]

This link you once provided may have been excerpted from a class using a John Stillwell book on the history of mathematics, which I understand has a large section on abstract algebra.

http://www2.math.ou.edu/~kmartin/nti/chap11.pdf

[desiresjab]

Though I am looking right at the examples in all these articles, I cannot yet see exactly how unique factorization has been recovered through ideals. I may be close, but not quite there yet. For me it may be a matter of tying together those definitions of ideals which depend on addition and those for muiltiplication.

When I get as close as I sense I am now, I begin to believe the job will be completed. I think we shall lay ideals bare soon.

[desiresjab]

I have to travel for a few days. Right when I would rather stay home and study I am forced to go on the road.

The difference between primes and irreducibles is still a problem to untangle. The article linked to last says the key idea in working with ideals is that the irreducibles and the primes do not match up in that realm. Now that is in the article almost word for word. As is the fact that the sum of two ideals is their GCD. Those are their words not mine. You know what I'm sane?

[YesNo]

Though I am looking right at the examples in all these articles, I cannot yet see exactly how unique factorization has been recovered through ideals. I may be close, but not quite there yet. For me it may be a matter of tying together those definitions of ideals which depend on addition and those for muiltiplication.


I am trying to make sense of that also, but I get distracted during the day.

I agree with what you said about adding terms from both <2> and <1+i> to get the ideal <2, 1+i>. In this case the ideal is principal and can be written as <1+i> in the Gaussian integers.

If one looks at Z[sqrt(-3)] there is an ideal that looks similar: <2, 1+sqrt(-3)>. Note that instead of i, we have sqrt(-3). This ideal is not principal because 2 does not divide into 1+sqrt(-3) like 2 was able to divide into 1+i in the Gaussian integers and get a Gaussian integer back. So what I understand we are to do is consider the ideal <2, 1+sqrt(-3)> as a new ideal number that we will add to Z[sqrt(-3)]. That is where the ideals help. That's how I understand it at the moment, but I might change my mind. This approach doesn't help with Z[sqrt(-5)] and that is where I am puzzled at the moment.

[desiresjab]

I am trying to make sense of that also, but I get distracted during the day.

I agree with what you said about adding terms from both <2> and <1+i> to get the ideal <2, 1+i>. In this case the ideal is principal and can be written as <1+i> in the Gaussian integers.

If one looks at Z[sqrt(-3)] there is an ideal that looks similar: <2, 1+sqrt(-3)>. Note that instead of i, we have sqrt(-3). This ideal is not principal because 2 does not divide into 1+sqrt(-3) like 2 was able to divide into 1+i in the Gaussian integers and get a Gaussian integer back. So what I understand we are to do is consider the ideal <2, 1+sqrt(-3)> as a new ideal number that we will add to Z[sqrt(-3)]. That is where the ideals help. That's how I understand it at the moment, but I might change my mind. This approach doesn't help with Z[sqrt(-5)] and that is where I am puzzled at the moment.

Of course the notion of 2 dividing into 1+i is odd anyway. It doesn't fit with our intuition. We could much easier appreciate how 1+i divides into 2. But with the peculiar definition of division in ideals we have 2 dividing 1+i.

I think better with my computer nearby. I will take a couple of days to try and bring everything together.

[YesNo]

I think I stated that wrong. 1+i divides 2 since (1+i)(1-i)=2. However, the ideal <2, 1+sqrt(-3)>in Z[sqrt(-3)] is not principal but it is prime or irreducible.

Using non-principal, but irreducible ideals is the way that ideals recovered unique factorization in those Dedekind domains that otherwise did not have unique factorization. There's something called the Fundamental Theorem of Ideal Theory that states that those Dedekind domains have unique factorization using ideals. I am trying to understand that proof. The Birkoff and MacLane book did not cover the proof and so I am reading Harry Pollard's "The Theory of Algebraic Numbers" to try to understand it better.

[desiresjab]

What I am finding is that whether a term is irreducible or prime seems to depend on the domain itself. For instance, in Z there does not seem to be any difference between irreducible and prime elements. One must always be aware if they are in an I.D, a P.I.D or a field to know these things. As I understand it, the definition of prime as we are familiar with it, is exactly the definition that suffices for irreducibles in ideal theory, but our familiar definition of primes is not sufficient for ideals.

There is also a theorem of Hilbert which converts any non-principal ideal into a principal one through multiplication of the ideal by a special Hilbert number. No details on this one yet.

Well, one thing we can say for sure is that a maximal ideal is always a prime ideal. Maximal ideals are fairly easy to get a handle on, thank Gog. I am unable to determine if maximal ideals include all of the prime ideals. I do not think so. So far, I believe there are other prime ideals which are not maximal but, of course, no maximal ideals which are not prime.

Sounds like you are really digging into the subject now. Some major insights must be on their way to you.

[YesNo]

Whatever insights I am getting have been discovered long ago. I am just sorting through the puzzle. I find this wikipedia page on Dedekind domains interesting at the moment: https://en.wikipedia.org/wiki/Dedekind_domain

Dedekind domains are integral domains in which there is unique factorization of ideals even though there may not be unique factorization of elements themselves. What that suggests to me is that there should be an example of a ring that is not a Dedekind domain, that is, where unique factorization of ideals does not work.

[YesNo]

Many insights are obvious after they are discovered. Then one wonders why one couldn't see them before. Here's one obvious insight that just recently became obvious to me.

Consider the greatest common divisor, g, of two integers, a and b. Given g, one can find two other integers x and y such that g = ax + by. Note the linear combination of a and b. If one considers all such combinations of a and b one has an ideal generated by a and b or <a, b>. That ideal is not principal, but because the integers are a principal ideal domain, one can find a single generator for <a, b> which would be g forming the principal ideal <g>.

I found on my bookshelves a translation of Dedekind's "Theory of Algebraic Integers" translated by John Stillwell. That insight about the gcd I mentioned above came from Stillwell's introduction, page 7. I forgot I even had that book and now for the first time, thanks to your discussion of these issues, desiresjab, I might actually finish reading it.

[desiresjab]

This is great. I urgently need all my concentration now to take steps in quicksand.

I believe Stillwell is the guy whose book (which you may be reading) was being used as the trext for the course taught here:

http://www2.math.ou.edu/~kmartin/nti/chap11.pdf

I am amazed by how many details I can pick up without the whole thing falling into place before my eyes. Sometimes even stubborn illusions are replaced by local understanding unable to force global epiphany.

Quite a few hurdles are terminological. When one finally receives the right information in the right form, the information usually sticks to the term like glue in the mind, but it often takes a long time for that event to happen--too long, for my tastes. I am dissatisfied with my mind's ability to gobble up these ideas like a young hen picking up corn. I would rather have a greater mind to work with than this old clunker.

* * * * *

There are three or four issues which if I could resolve, I would have a decent grip on ideals, unique factorization and field extensions, and precisely how they all fit together and what in concert they have acheived.

Another barricade is that much of ideal theory is expressed in group theoretical notation with groups leading the idea train. From the beginning ideals had deep connections with group theory and was developed with them in mind. I do not know that subject well enough to catch all the hints.

[YesNo]

The way I avoid quicksand is to sleep on it, stop reading, skim or switch to some other text when the one I'm reading becomes too difficult. Of course, that means I might never go back to the original text which is what happened with that Dedekind book long ago. I have only read Stillwell's introduction, but he has a very good style. The parts I understand are clear. The parts I don't understand it is probably my fault that I don't understand them.

Another insight that is now obvious to me is that when one extends the rationals Q with sqrt(-1) = i to get Q(i) one gets a field smaller than the complex numbers. It doesn't even contain all the real numbers. For example, it doesn't include any transcendental numbers such as pi or e because they aren't in Q and they can't be formed from a + bi where a and b are in Q. But when I see something written as a + bi I assume I am working in the complex numbers when actually I am working in a subfield of the complex numbers.

[desiresjab]

The only number in the whole field which is not rational is the field extension itself. But I must confess I am at a total loss understanding the difference between extending to sqrt (-3) and extending by √-3i or something like that. I truly do not understand this difference or whatever advantage might accrue to it. This issue is one of the three or four items I still must bring under control. You addressed it once but I still could not understand.

[desiresjab]

Another thing which still throws me is--it is just like me these days to forget what it was before I have the sentence finished, dammit!

Oh, the other thing which confuses me is how 1+√-3 et al got into the picture at all. We were considering √-3, for instance, and suddenly here we are with 1+√-3. What justifies that 1 out in front? I understand it helps because it has a conjugate and I am not certain if mere √-3 does. Still, I have a hard time justifying its presence. Do you understand what the justification is?

[desiresjab]

Suddenly, I think I see how a Quotient Ring works.

Take the polynomial x2+3=0

If we subtract x2+3 from any other polynomial, the difference either is or is not a multiple of x2+3. If it is a multiple, we call that result equivalent to x2+3 itself. We may even call it equivalent to the second polynomial. I am not sure about that point. Probably not, actually. Well, hmmm..., I don't know. More thought.

This is really only modular arithmetic using x2+3 as the modulus.

[desiresjab]

If I take (x2+4)-(x2+3)=1,

that seems to imply that to me that 1 and x2+3 are equivalent, since it does seem that everything (and this includes x2+3) is a multiple of 1.

Yet it definitely puts x2+4 in the residue class (equivalence class?) of 1, where x2+3 is in the 0 class mod itself, so it is hard for me to see them as equivalent.

The above indicates to me that 1 and x2+4 are in the same residue class (equivalence class?), not that 1 and x2+3 are, which is impossible when x2+3 is the modulus.

It would be possible for polynomials A and B to be in the same residue class. In this case their difference would be, too. A, B and B-A would all have the same residue class.

Each residue class (equivalence class?) must form an ideal. All polynomials leaving a residue of 1, for instance, would form an ideal. All polynomials leaving a residue of 2 would form another ideal, etc. All polynomials leaving a residue of technically 0 would be in the same class as x2+3.

[YesNo]

Another thing which still throws me is--it is just like me these days to forget what it was before I have the sentence finished, dammit!

Oh, the other thing which confuses me is how 1+√-3 et al got into the picture at all. We were considering √-3, for instance, and suddenly here we are with 1+√-3. What justifies that 1 out in front? I understand it helps because it has a conjugate and I am not certain if mere √-3 does. Still, I have a hard time justifying its presence. Do you understand what the justification is?

I think sqrt(-3) = sqrt(-1)*sqrt(3) = i*sqrt(3).

The a + b*sqrt(-3) has two terms because it is generated by 1 and sqrt(-3) over the rationals. So one has a linear combination of both generators. If we let a and b be in Q, then a*1 + b*sqrt(-3) can take any value in the field extension Q(sqrt(-3)).

[YesNo]

If I take (x2+4)-(x2+3)=1,

that seems to imply that to me that 1 and x2+3 are equivalent, since it does seem that everything (and this includes x2+3) is a multiple of 1.

Yet it definitely puts x2+4 in the residue class (equivalence class?) of 1, where x2+3 is in the 0 class mod itself, so it is hard for me to see them as equivalent.

The above indicates to me that 1 and x2+4 are in the same residue class (equivalence class?), not that 1 and x2+3 are, which is impossible when x2+3 is the modulus.

It would be possible for polynomials A and B to be in the same residue class. In this case their difference would be, too. A, B and B-A would all have the same residue class.

Each residue class (equivalence class?) must form an ideal. All polynomials leaving a residue of 1, for instance, would form an ideal. All polynomials leaving a residue of 2 would form another ideal, etc. All polynomials leaving a residue of technically 0 would be in the same class as x2+3.

The equation at the top is an identity, so it should work for any x. However a polynomial is 0 for only a few roots.

If x2+3 is the modulus then any polynomial times that one have those two roots in common. It is like a prime, say 3. Any integer times 3 would be in the ideal generated by 3.

[desiresjab]

Okay. I am back. I think I see everything so far except the fundamental difference between √-5 and √-3 and why a different treatment is necessary, since we were missing the treatment of √-3 in the article we have linked to several times. That excerpt speaks of that treatment but never shows it, though it gives a hint for the enlightened.

[YesNo]

As I see it now, I don't think the treatment from the perspective of ideals is different. In both cases, when we look at ideals the ideals factor into prime ideals (although those prime ideas are not principal ideals because they have more than one generator). What makes a prime is the property, going back to Euclid, that if a prime p (from whatever ring) divides a product, ab, then p must divide either a or p must divide b. The prime is not allowed to divide some of each as 6 dividing 2*3 does.

When we look at elements of the ring of integers they don't factor into primes. I still don't understand the fundamental theorem of ideal theory so I don't see why unique factorization has to work yet. The examples showing that it does work can be checked, but they are just examples.

There is that other difference in the original article we looked at. I think it referred to extending the Z[sqrt(-3)] ring, however, I don't see it at the moment.

[desiresjab]

On the difference between treatments of 3 and 5 in the second paragraph of the article, I notice that 5 can be factored in Gaussian integers without the use of a √ sign, and that 3 cannot be, and I wonder if this has any bearing on their primary difference, which I suppose must ultimately be related to the fact that 3 is a Gaussian prime and 5 is not.

We also know that Gaussian primes cannot be represented as the sum of two squares. That means 4n+3 type primes in the integers.

What I am still trying to figure out is what the article means when they say that the quotient field of Z[√5] already contains all the integers under it, implying that the quotient field of Z[√3] does not. And then the article says that is why they cannot simply add more numbers to the extension field of √5 they way they did for √3. Of course that mysterious work comes in a previous part of the book we did not get to see.

Could it be that the failure of 3 to factor nicely without stepping out of Gaussian integers the way 5 does into (2+i)(2-i) is involved?

I am sure there is a simple division process to demonstrate this. I am not sure how to do it. But I do have a crude idea.

If the √ of anything is part of the quotient field, I see how dividing by it could mess things up. Not too helpful, I know, but a small insight. You would not get nice integers anymore.

It hadn't ought to be that hard to construct quotient fields for 3 and 5, should it, and demonstrate how the latter already contains all the integers under it and the former does not?

[YesNo]

I am having trouble understand that same distinction mentioned in the article. Dedekind introduced the language of ideals, but Kummer came before him and introduced ideal numbers. I think they are referring to Kummer's approach rather than Dedekind's, but I don't know yet. The sqrt(-5) is supposedly the critical one leading to the results in algebraic number theory, not sqrt(-3), but I don't see it yet.