Cosmology

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])

Quote:

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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.