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Feynman on Fermat's Last Theorem

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Re: Feynman on Fermat's Last Theorem

#51

Post by conistonwater »

acqq wrote: post #48
conistonwater wrote: post #29

1e+33 is not such a big number as far as number theory goes (for example: http://mathoverflow.net/questions/15444/examples-of-eventual... , https://en.wikipedia.org/wiki/Skewes%27_number ). It's still a nice exercise. As another example, consider the question of whether there exists a right-angled triangle with rational sides, having an area of 157.

If I understand it right, it's a number with which he could have estimated if it's worth using a computer to perform a brute force search for a possible solution. Once he had 1e-33 for all n > 100 that could mean that even trying with 1M computers where each tries 10G solutions per second (1e6*1e10) some millions of years could pass without the positive result. Then it's exactly reasonable to say "for my money Fermat…

I can't believe that the right interpretation of what he said. He must have known what the actual FLT meant. Furthermore, it would go against the mathematical tradition of what it means to "think that a conjecture is true". Compare with RH, for example: when people say it's probably true they absolutely do not mean it's true for all small numbers, in part because there are built-up areas of mathematics that depend on it being exactly true. Besides, brute-force search is a pretty terrible algorithm (in general), so finding out that it fails on a particular problem isn't that interesting.

Re: Feynman on Fermat's Last Theorem

#52

Post by acqq »

conistonwater wrote: post #51
acqq wrote: post #48

Earlier quoted context omitted.

If I understand it right, it's a number with which he could have estimated if it's worth using a computer to perform a brute force search for a possible solution. Once he had 1e-33 for all n > 100 that could mean that even trying with 1M computers where each tries 10G solutions per second (1e6*1e10) some millions of years could pass without the positive result. Then it's exactly reasonable to say "for my money Fermat…

I can't believe that the right interpretation of what he said. He must have known what the actual FLT meant. Furthermore, it would go against the mathematical tradition of what it means to "think that a conjecture is true". Compare with RH, for example: when people say it's probably true they absolutely do not mean it's true for all small numbers, in part because there are built-up areas of mathematics that depend on…

He died in 1988.

Imagine somebody came to you at these "early" times (Wiles proved the theorem in 1995) with a "grand project" to use a lot of computers to search for a possible "solution," being able to try 10 billion Ns in one second. What would be your argument against such a project? Would you try some similar derivation to get an estimate of success?

I know, brute forcing all integers is impossible, but you can even "imagine" a "superquantum (and now not existing) computer" which can do a lot of small integers in parallel.

And yes, I know, probabilities aren't proofs, especially not for integers. One counterexample is enough:

3987^{12} + 4365^{12} = 4472^{12}

Also interesting to see the accidental(?) order of magnitude of the numbers involved.

Re: Feynman on Fermat's Last Theorem

#53

Post by lsh123 »

ambrop7 wrote: post #26
lsh123 wrote: post #21

Earlier quoted context omitted.

The meaning here is: pick a positive integer N, what is the chance (aka probability) of it being an n-th power of another positive integer. And you are correct, this probability depends on N.

I don't get it. The probability must depend on how likely I am to select any specific integer, i.e. probability mass function of N. The probability cannot depend on the value of the random variable itself but could involve any parameters that define its distribution. I would consider something like the following a valid question: "Let N be a random integer between 0 and M-1 with uniform distribution. What is the prob…

Imagine you don't have an easy way to solve the equation x^n = N where x is an integer. Yet, for a given N you want to find a probability that such x (solution) exists. There is nothing here about how you select N.

Re: Feynman on Fermat's Last Theorem

#54

Post by qmalzp »

sajid wrote: post #15

If FLT hadn't already been proven then Feynman's argument would explain away the substantial numerical evidence collected in it's favor. In other words, it would suggest that FLT is just a statistical accident and true for no particular reason. Of course, we now know that FLT is related to deep ideas in number theory.

I mean Feynman's argument really seems to show that it should hold for sufficiently large N. I think one could reasonably argue that it's a statistical accident that FLT holds for N >= 3 as opposed to say, N >= 100. I think even Wiles' proof only works for N sufficiently large (at least N >= 5). Small N of course were handled by earlier results.

Re: Feynman on Fermat's Last Theorem

#56

Post by thaumasiotes »

Someone wrote: post #7
joshdick wrote: post #2

I wonder how many false conjectures could pass muster using this sort of probabilistic argument.

I guess every false conjecture can be made to pass it. The trick is to make the set of items searched in large enough. For example, to show that no elephants exist, start with the (infinite) set of all possible chromosome sets. The proportion of them that produces an elephant is zero. QED. Examples from mathematics: The number 42 does not exist (logic: pick an integer. The probability that it equals 42 is zero. QED)…

> to show that no elephants exist, start with the (infinite) set of all possible chromosome sets. The proportion of them that produces an elephant is zero

This is not obvious. If you're going to postulate an infinite number of possible chromosome sets, you're also going to have to admit that an infinite subset of them all produce elephants.

For example, if you show genome A which does not produce an elephant, perhaps I could show genome A', consisting of (1) genome A; (2) a reference elephant genome; (3) some chromosomes that have the effect of disabling genome A.

Re: Feynman on Fermat's Last Theorem

#57

Post by spott »

ambrop7 wrote: post #26
lsh123 wrote: post #21

Earlier quoted context omitted.

The meaning here is: pick a positive integer N, what is the chance (aka probability) of it being an n-th power of another positive integer. And you are correct, this probability depends on N.

I don't get it. The probability must depend on how likely I am to select any specific integer, i.e. probability mass function of N. The probability cannot depend on the value of the random variable itself but could involve any parameters that define its distribution. I would consider something like the following a valid question: "Let N be a random integer between 0 and M-1 with uniform distribution. What is the prob…

You are thinking "given N, what is the probability that N is a perfect nth power?". Think about it in the reverse. "given N, what is the probability that the nth root of N is an integer". For a whole bunch of different Ns, we will get some real number between 0 and 1.

To match up to your example. "For a number M, m is the remainder of M: m = M - floor(M), what is the probability that m is zero given M"

Re: Feynman on Fermat's Last Theorem

#58

Post by gohrt »

ambrop7 wrote: post #17

"the probability that N is a perfect n^nth power..." Can someone explain what probability means here in relation to N? From my understanding, it depends on what N is for you. If it's a constant, that probability is obviously 0 or 1. So that can't be it. Then N must be some kind of random variable. But with what distribution? And in what kind of system can the probability of event(random_variable) involve random_varia…

It's a simple density in the integers.

Either globally: Proposition P is true for n of the numbers from 1:N -> p(P,N) = n/N

or locally: Proposition P is true for n of the numbers from N-k to N+k -> p(P, N, k) = n/k

and pick a reasonable k (where p(P, N, K) is relatively stable for a neighborhood around k)

Re: Feynman on Fermat's Last Theorem

#59

Post by talaketu »

conistonwater wrote: post #29

1e+33 is not such a big number as far as number theory goes (for example: http://mathoverflow.net/questions/15444/examples-of-eventual... , https://en.wikipedia.org/wiki/Skewes%27_number ). It's still a nice exercise. As another example, consider the question of whether there exists a right-angled triangle with rational sides, having an area of 157.

Wasn't the idea that he integrated on n for 100 to infinity, to arrive at that probability? So the probably should be interpreted as the probability of an eventual counter-example.
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