Cosmology

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/nu...er_of_divisors

Lol! Some "f" spam here.

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.

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

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Originally Posted by 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.

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

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Originally Posted by Danik 2016

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

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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.

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.

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.

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.

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.

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.

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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.

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%...tient_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.

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Originally Posted by 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%...tient_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.

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They call that a cototient, eh?

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.

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.

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Originally Posted by 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.

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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.

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Originally Posted by 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.

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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?

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 1^n-1=1 mod n and (n-1)^n-1 = -1^n-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

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Originally Posted by 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 1^n-1=1 mod n and (n-1)^n-1 = -1^n-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

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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, ...q_k}) 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.

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

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Originally Posted by 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.

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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.

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Originally Posted by YesNo

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

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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?

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])

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Originally Posted by 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])

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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.

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.

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.

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

Continued fractions also have repetitions that might be useful.

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Originally Posted by YesNo

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

Continued fractions also have repetitions that might be useful.

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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.

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.

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 e^iπ+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.

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.

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.

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Originally Posted by 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.

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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 10^p-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).

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.

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.

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.

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.

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 O_K. 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.

Which Wikipedia article were you reading?

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Originally Posted by YesNo

Which Wikipedia article were you reading?

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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/Fundam..._homomorphisms

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

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