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Feynman on Fermat's Last Theorem

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Re: Feynman on Fermat's Last Theorem

#11

Post by nappy-doo »

jakeva wrote: post #10
Someone wrote: post #3

Using way simpler math, the probability that an even number is prime is zero. One cannot conclude from that that there are no even primes (in fact, many mathematicians think the smallest prime is even ( https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.h... )) Mathematicians do use this kind of back of the envelope calculations to get a feeling for whether a statement may be true, but they can never prove so…

Your link doesn't work for me…

Remove trailing ):

https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.h...

Re: Feynman on Fermat's Last Theorem

#12

Post by coherentpony »

Someone wrote: post #3

Using way simpler math, the probability that an even number is prime is zero. One cannot conclude from that that there are no even primes (in fact, many mathematicians think the smallest prime is even ( https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.h... )) Mathematicians do use this kind of back of the envelope calculations to get a feeling for whether a statement may be true, but they can never prove so…

> in fact, many mathematicians think the smallest prime is even

Up to sign? The smallest prime is even.

Re: Feynman on Fermat's Last Theorem

#13

Post by Retra »

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 encoded in the number than there is necessary to solve the problem, as which point no larger number could be a solution.

This is basically how induction works anyway: you produce a base case and an algorithm, and infer that the information contained therein meets the needs of the problem. Then any number which would encode more information is irrelevant (and thus sufficient) and you have an inductive proof.

Re: Feynman on Fermat's Last Theorem

#14

Post by stouset »

Someone wrote: post #3

Using way simpler math, the probability that an even number is prime is zero. One cannot conclude from that that there are no even primes (in fact, many mathematicians think the smallest prime is even ( https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.h... )) Mathematicians do use this kind of back of the envelope calculations to get a feeling for whether a statement may be true, but they can never prove so…

I don't quite understand your comment.

First, the smallest prime (2) is even.

Second, your first statement can substitute "an even number is prime" with "a multiple of n is prime" for all prime n, leading us to conclude that the odds of any multiple of 3, 5, 7, 11, 13, et al being prime are zero thus seemingly statistically disproving the existence of any prime number which is absurd.

Re: Feynman on Fermat's Last Theorem

#15

Post by sajid »

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.

Re: Feynman on Fermat's Last Theorem

#16

Post by Someone »

jakeva wrote: post #10
Someone wrote: post #3

Using way simpler math, the probability that an even number is prime is zero. One cannot conclude from that that there are no even primes (in fact, many mathematicians think the smallest prime is even ( https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.h... )) Mathematicians do use this kind of back of the envelope calculations to get a feeling for whether a statement may be true, but they can never prove so…

Your link doesn't work for me…

HN's URL detector doesn't handle nested parentheses, it seems. I added a space to fix this.

Re: Feynman on Fermat's Last Theorem

#17

Post by ambrop7 »

"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_variable itself?

Re: Feynman on Fermat's Last Theorem

#18

Post by Someone »

coherentpony wrote: post #12
Someone wrote: post #3

Using way simpler math, the probability that an even number is prime is zero. One cannot conclude from that that there are no even primes (in fact, many mathematicians think the smallest prime is even ( https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.h... )) Mathematicians do use this kind of back of the envelope calculations to get a feeling for whether a statement may be true, but they can never prove so…

> 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.

Re: Feynman on Fermat's Last Theorem

#19

Post by aji »

CJefferson wrote: post #9
Someone wrote: post #7

Earlier 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)…

Several of these don't work, as it's assumed you tackle smaller values in another way, before your -> infinity method kicks in, so most of these would be easily knocked off.

not sure I follow

"the integers don't exist, because the percent of real numbers that are integers is effectively 0" is what the parent post is implying, which "works", in the sense that using this proof methodology "works"

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