I don't understand why the distance between root(n, N) and root(n, N+1) is the probability that N is a perfect nth power. What if we choose to compute the distance between root(n, N) and root(n, N+0.5) ? By the same logic, if N is a perfect nth power, there exists one integer in the interval [root(n, N),root(n, N+0.5)], and since the distance between all consecutive integers is one, the probability is the ratio etc ... And what about 0.2? 0.1? We can keep going until the probability becomes zero ! And this can't be !
Feynman on Fermat's Last Theorem
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Re: Feynman on Fermat's Last Theorem
#62It's a nice exercise However Number Theory is a different beast altogether It has as much to do with "regular" math as English and Latin have in common, even though they are written with the same alphabet.
Are you saying that number theory has not in fact been a core part of mathematics for hundreds of years.
Re: Feynman on Fermat's Last Theorem
#63> Feynman also knew about Sophie Germain’s result, who proved in the early 19th century that Fermat’s equation has no solution for n≤100.
That's intriguing! What's special about the number 100 that you can prove the case for n≤100?
Re: Feynman on Fermat's Last Theorem
#64> Feynman also knew about Sophie Germain’s result, who proved in the early 19th century that Fermat’s equation has no solution for n≤100. That's intriguing! What's special about the number 100 that you can prove the case for n≤100?
That description of what Sophie Germain did is slightly glib and inaccurate, I believe. I will give an abbreviated, but accurate, to the best of my knowledge, explanation of what happened:
Note that, if you have proven Fermat's Little Theorem for some exponent, then you have also proven it for every multiple of that exponent. Thus, to prove it for all exponents (> 2) up to some limit, it suffices to prove it for all odd prime exponents up to that limit, as well as for exponent 4. Fermat himself gave an argument which worked for exponent 4, so afterwards, one could consider only odd prime exponents.
Note also that, if p is prime, and we have a solution to x^p + y^p = z^p, then out of {x, y, z}, precisely 0, 1, or all 3 are divisible by p (i.e., if any two were divisible by p, then so would be the third). If indeed all 3 are divisible by p, we may accordingly divide all through by p to obtain a smaller solution; thus, if there is any solution, there is a minimal solution where precisely 0 or 1 of {x, y, z} are divisible by p.
So to prove Fermat's Little Theorem for prime exponent p, it suffices to prove both of the following claims for all solutions to x^p + y^p = z^p:
(A) It cannot be the case that precisely 0 of {x, y, z} are divisible by p
(B) It cannot be the case that precisely 1 of {x, y, z} is divisible by p
If this can be done for every odd prime p up to some limit, FLT is established for all exponents up to that limit.
Germain did not do this. She did not have a strategy for proving (B). This prevented her from establishing FLT in full for any exponent.
Germain did manage, however, to discover a strategy for establishing (A) for various p. Specifically, she discovered a sufficient condition for (A) was the existence of another prime t in a certain decidable relation to p. Germain then manually searched for and discovered such t for each prime p Why'd she stop there? Because it seemed like a nice place to stop. Nothing special about 100 except the human factor. One could keep going, and indeed, Legendre extended Germain's results to each prime I may still not have gotten the story exactly correct, or noted all the pertinent details, but for more, see https://www.agnesscott.edu/lriddle/women/germain-FLT/SGandFL..., on which I based the above description.
Re: Feynman on Fermat's Last Theorem
#65[deleted]
Re: Feynman on Fermat's Last Theorem
#66Very cute argument, and very much in his style--he was famous, as the author notes, for heuristic arguments that weren't very formalizable but had a lot of beauty. One story I heard is that a computer scientist tried to explain the P=NP? problem to him; Feymnan couldn't understand why this was a problem. It was obviously true that P != NP, what even needed proving?
Per "Surely You're Joking" and the article, one of the cognitive techniques Feynman used was to keep a physical example, a demo, in his mind that conformed to the math being explained. I wonder if he used that for this, and what his model was.
If you imagine it as a logical statement, represented physically in writing: P ~P
Then, I think it is clear the statement is not true. NP is trivially definitionally not equivalent (equal) to P.
Re: Feynman on Fermat's Last Theorem
#67I have never seen a probabilistic approach to FLT but there is a branch of math called Probabilistic Number Theory
https://people.math.ethz.ch/~kowalski/probabilistic-number-t...
I am also not 100% sure Feynman wrote something like this. However the 2014 Fields Medal was awarded to Manjul Bhargava for studying random elliptic curves.
https://www.quantamagazine.org/20140812-the-musical-magical-...
Re: Feynman on Fermat's Last Theorem
#68Earlier quoted context omitted.
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)…
Whoa, hang on. Statistical arguments for unproven conjectures are bad, but this counterargument is as bad or worse, especially when you start talking about infinity. Just to address your first example: > The number 42 does not exist (logic: pick an integer. The probability that it equals 42 is zero. QED) I object! What is your probability distribution function over the integers? Your phrasing sort of implies a unifor…
It is easy to address your objection. On a uniform distribution on [n] := {0, 1, 2, ...n}, P(X=42)->0 as n->infty. This is similar to Feynmann's argument.
Re: Feynman on Fermat's Last Theorem
#69There's a similar argument for why Goldbach's conjecture* might just be a statistical fluke. I remember realizing this in high school and deriving the probability – alas it's not a novel idea and Wikipedia has a great summary: https://en.wikipedia.org/wiki/Goldbach%27s_conjecture#Heuris...
It's very possible that it's an unprovable true conjecture, which is kind of interesting in itself.
* every even number can be written as a sum of two primes
Re: Feynman on Fermat's Last Theorem
#70Earlier quoted context omitted.
Per "Surely You're Joking" and the article, one of the cognitive techniques Feynman used was to keep a physical example, a demo, in his mind that conformed to the math being explained. I wonder if he used that for this, and what his model was.
If you imagine it as a logical statement, represented physically in writing: P ~P Then, I think it is clear the statement is not true. NP is trivially definitionally not equivalent (equal) to P.
NP is most certainly not "trivially definitionally not equivalent (equal) to P". (Do you think everyone puzzling over the question of whether P = NP is a moron? You are aware there is a million dollar prize for a proof one way or the other, yes?). What definition are you using for NP?