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Feynman on Fermat's Last Theorem (2016)

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Re: Feynman on Fermat's Last Theorem (2016)

#11

Post by jordigh »

Sigh, this proof bugs me so much.

Compute an approximate distance between nth powers, interpret this as the probability of an integer being an nth power, integrate this probability over the sum x^n + y^n, see that the probability of this being an nth power is also very low.

I guess this is close enough for government work, but it's so utterly fallacious. For example, the distance between n^2 and (n-1)^2 is 2n - 1. That "means" that the "probability" of N being a perfect square is about 1/(sqrt(2N - 1)). This probability also goes to zero in the limit as N goes to infinity. Not very quickly, but it does.

Does that mean that square numbers don't exist?

We have many examples of conjectures being disproved by very large counterexamples:

https://www.quora.com/What-is-an-example-of-a-conjecture-tha...

Re: Feynman on Fermat's Last Theorem (2016)

#12

Post by ykler »

jordigh wrote: post #11

Sigh, this proof bugs me so much. Compute an approximate distance between nth powers, interpret this as the probability of an integer being an nth power, integrate this probability over the sum x^n + y^n, see that the probability of this being an nth power is also very low. I guess this is close enough for government work, but it's so utterly fallacious. For example, the distance between n^2 and (n-1)^2 is 2n - 1. Th…

You forgot to sum over N

Re: Feynman on Fermat's Last Theorem (2016)

#13

Post by throwanem »

auggierose wrote: post #8
throwanem wrote: post #6

Earlier quoted context omitted.

> the job of practical engineering is to provide reliable, correct solutions in a timely, in-budget manner Sure. And I'm sure you'll agree this job is easier to do in direct proportion to the celerity with which we eliminate impractical approaches.

You can eliminate impractical approaches all day long. At some point you have to start with the practical approach that you know will work a-priori.

When you have the option, sure! Not all problems are so tractable.

Re: Feynman on Fermat's Last Theorem (2016)

#14

Post by jordigh »

ykler wrote: post #12
jordigh wrote: post #11

Sigh, this proof bugs me so much. Compute an approximate distance between nth powers, interpret this as the probability of an integer being an nth power, integrate this probability over the sum x^n + y^n, see that the probability of this being an nth power is also very low. I guess this is close enough for government work, but it's so utterly fallacious. For example, the distance between n^2 and (n-1)^2 is 2n - 1. Th…

You forgot to sum over N

Yeah, okay, I messed up. Let's argue in a different way: there are approximately sqrt(N) squares between 1 and N, so let's call sqrt(N)/N = 1/sqrt(N) the "probability" that N itself is square. That also goes to zero as N goes to infinity. If we pick higher powers of N, it goes to zero much quicker.

Re: Feynman on Fermat's Last Theorem (2016)

#15

Post by todd8 »

jordigh wrote: post #11

Sigh, this proof bugs me so much. Compute an approximate distance between nth powers, interpret this as the probability of an integer being an nth power, integrate this probability over the sum x^n + y^n, see that the probability of this being an nth power is also very low. I guess this is close enough for government work, but it's so utterly fallacious. For example, the distance between n^2 and (n-1)^2 is 2n - 1. Th…

Interesting, and startling to me when I first learned of it, is the probabilistic method [1], now not infrequently used in combinatorics to prove the existence of a solution to some question without constructing it. One proves that the probability of existence is strictly greater than zero.

[1] https://www.amazon.com/Probabilistic-Method-Discrete-Mathema...

Re: Feynman on Fermat's Last Theorem (2016)

#16

Post by jordigh »

todd8 wrote: post #15
jordigh wrote: post #11

Sigh, this proof bugs me so much. Compute an approximate distance between nth powers, interpret this as the probability of an integer being an nth power, integrate this probability over the sum x^n + y^n, see that the probability of this being an nth power is also very low. I guess this is close enough for government work, but it's so utterly fallacious. For example, the distance between n^2 and (n-1)^2 is 2n - 1. Th…

Interesting, and startling to me when I first learned of it, is the probabilistic method [1], now not infrequently used in combinatorics to prove the existence of a solution to some question without constructing it. One proves that the probability of existence is strictly greater than zero. [1] https://www.amazon.com/Probabilistic-Method-Discrete-Mathema...

Yeah, that's really different, though. It's essentially a way to simplify counting. But the "converse" of the probabilistic method doesn't work: zero probability doesn't mean inexistence in an infinite set.

Re: Feynman on Fermat's Last Theorem (2016)

#17

Post by pmiller2 »

jordigh wrote: post #11

Sigh, this proof bugs me so much. Compute an approximate distance between nth powers, interpret this as the probability of an integer being an nth power, integrate this probability over the sum x^n + y^n, see that the probability of this being an nth power is also very low. I guess this is close enough for government work, but it's so utterly fallacious. For example, the distance between n^2 and (n-1)^2 is 2n - 1. Th…

I have to agree that this argument is utterly unconvincing, even as a reason to believe FLT may be true intuitively. It simply dismisses with a wink and a nod the idea that there might be one single counterexample out there for one single exponent (or, a finite number of counterexamples for a finite number of exponents). In other words, it does nothing to argue against the set of counterexamples being of measure 0. Indeed, until Wiles finally proved FLT, the relative lack of progress on Siegel's conjecture on the infinitude of regular primes would tend to lend support to the view that a counterexample might exist somewhere out there.

Re: Feynman on Fermat's Last Theorem (2016)

#18

Post by nshepperd »

jordigh wrote: post #14
ykler wrote: post #12

Earlier quoted context omitted.

You forgot to sum over N

Yeah, okay, I messed up. Let's argue in a different way: there are approximately sqrt(N) squares between 1 and N, so let's call sqrt(N)/N = 1/sqrt(N) the "probability" that N itself is square. That also goes to zero as N goes to infinity. If we pick higher powers of N, it goes to zero much quicker.

That's not what ykler meant. 1/sqrt(N) is the "probability" that N itself is square. We don't care about that, we want the probability that there is any solution to x^2 = N, which is sum{N=0..∞} (1/sqrt(N)). That does not go to zero.

ETA: Well, actually the above is the expected number of solutions, so naturally it diverges because there are (infinite) solutions. A more proper way would be to calculate the probability that there are no solutions, which indeed goes to zero. But the probability of there being a solution, and the expected number of solutions, are the same if it is << 1.

Re: Feynman on Fermat's Last Theorem (2016)

#19

Post by auggierose »

throwanem wrote: post #13
auggierose wrote: post #8

Earlier quoted context omitted.

You can eliminate impractical approaches all day long. At some point you have to start with the practical approach that you know will work a-priori.

When you have the option, sure! Not all problems are so tractable.

Granted. Still, I think there is a fundamental difference: In theoretical physics, a good theory of how things work (or how they definitely don't work) is enough as the result of your work, you don't need a working product in the end. In practical engineering, you do!

Let me also add that I am serious about theoretical engineering, actually that's one of my favourite occupations, together with practical engineering.

Re: Feynman on Fermat's Last Theorem (2016)

#20

Post by alimw »

pmiller2 wrote: post #17
jordigh wrote: post #11

Sigh, this proof bugs me so much. Compute an approximate distance between nth powers, interpret this as the probability of an integer being an nth power, integrate this probability over the sum x^n + y^n, see that the probability of this being an nth power is also very low. I guess this is close enough for government work, but it's so utterly fallacious. For example, the distance between n^2 and (n-1)^2 is 2n - 1. Th…

I have to agree that this argument is utterly unconvincing, even as a reason to believe FLT may be true intuitively. It simply dismisses with a wink and a nod the idea that there might be one single counterexample out there for one single exponent (or, a finite number of counterexamples for a finite number of exponents). In other words, it does nothing to argue against the set of counterexamples being of measure 0. I…

I haven't read it closely, but it looks to me as if the calculation estimates the expected number of counterexamples rather than the "measure" of them (however you've chosen to define that).
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