Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Feynman on Fermat's Last Theorem

lbatalha.com

Post reply on HN
Print view View original

21–30 of 113 posts

Re: Feynman on Fermat's Last Theorem

#21

Post by lsh123 »

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…

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.

Re: Feynman on Fermat's Last Theorem

#25

Post by coherentpony »

Someone wrote: post #18
coherentpony wrote: post #12

Earlier quoted context omitted.

> in fact, many mathematicians think the smallest prime is even Up to sign? The smallest prime is even.

1 could be a prime, too, as for example Legendre, Lesbegue, Cayley (in the Encyclopædia Britannica), Kronecker, Hardy and Sagan stated at least once (see the URL I linked to earlier) One isn't typically called a prime for the same reason as mathematicians typically say 0^0 equals 1; it makes many theorems and proofs look better.

Hm. I'm a mathematician. I always thought it wasn't called prime because it breaks uniqueness for the fundamental theorem of arithmetic.

Re: Feynman on Fermat's Last Theorem

#26

Post by ambrop7 »

lsh123 wrote: post #21
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…

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 probability that N is even?"

Then an answer could be "the probability that N is even is 1/2 if M is even and (M+1)/(2M) if M is odd". See this does not involve N but does involve M which is a parameter for the distribution of N.

Your explanation seems to invoke some "common sense" which I am not able to unify with my understanding of probability theory.

Re: Feynman on Fermat's Last Theorem

#27

Post by kryptiskt »

Retra wrote: post #13
supergirl wrote: post #5

finding that the probability is extremely small doesn't really get you any closer to proving it. for such a hard to prove theorem, it makes sense the probability is small. that's why it was interesting in the first place. I find it hard to believe Feynman concluded from this that the theorem is probably correct. it only takes 1 case among an infinity to make the theorem false.

I think the idea here is that, in general, simple coherent properties of number tend to show themselves on smaller numbers more readily than on larger ones. It would be quite unusual to have a problem stated in such simple terms and require a solution so far away. Thus this probability is used to infer just how large such a number must be, and eventually you're going to hit a point where there is more information enc…

Littlewood's prime counting theorem is an example what can hide up among the big numbers (https://en.wikipedia.org/wiki/Skewes%27_number)

Re: Feynman on Fermat's Last Theorem

#28

Post by tgb »

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)…

Naw most of these wouldn't work when actually written out as Feynman did. Example: you can easily give an upper bound to the "chance that N is a prime" that goes to zero as N increases. But you would also need to show that it's sum from 0 to infinity over all N also goes to zero.

In fact, there's the classic result that this probability is about 1/log(N) [1], which diverges towards infinity. Hence you would probabilistically expect infinite primes and would be correct.

[1] https://en.wikipedia.org/wiki/Prime_number_theorem

Re: Feynman on Fermat's Last Theorem

#29

Post by conistonwater »

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.

Post reply on HN