Feynman on Fermat's Last Theorem • Hacker News Board

Re: Feynman on Fermat's Last Theorem

#81

Post by Houshalter » Sat, Jul 02, 2016, 1:33 PM UTC

erikbern wrote: post #69

There'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 wr…

What's interesting about this argument is, if its true, it's an example of a simple conjecture that is true but unprovable. But not like Godel sentences, which can be proved under different axioms, or proved to be unprovable. This would be completely unprovable. And not because of some logical paradox, but simply because there is no mathematical reason for it to be false, it's just a coincidence.

There are also probably many conjectures like this. Freeman Dyson constructed one, that no digits of powers of two, reversed, make a power of 5. The reasoning being that the digits grow big quickly, and the base 10 representation is basically random and uncorelared with it.

I think other conjectures about digits of numbers are similar. E.g. Mathematicians have found it impossible to prove that any numbers are normal, or even if pi contains infinite 5's, and many similar properties. These things may only be probabilistically true.

Re: Feynman on Fermat's Last Theorem

#82

Post by Kenji » Sat, Jul 02, 2016, 1:51 PM UTC

fareesh wrote: post #78

My math is passably college level, so most of these proofs go over my head. I've always thought of FLT as stating "You cannot make n-3 cubes of side z by stacking/tiling n-3 cubes of side x and y", at which point it becomes something of a tessellation problem in 3 dimensions, which would seem to me to be the intuitive way to go about proving this.

I like your approach. It's funny, I never thought about it that way, I never even considered visualizing it. I always looked at it from the perspective of pure number theory.

Re: Feynman on Fermat's Last Theorem

#83

Post by jordigh » Sat, Jul 02, 2016, 1:51 PM UTC

tgb wrote: post #80

jordigh wrote: post #68

Earlier quoted context omitted.

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.

You're forgetting to integrate the probability over the domain 0 to n. If you do, you always get 1 for n at least 42.

Those are two separate events. I'm talking about the event n = 42, which makes sense for all [n]. You're talking about the event n >= 42, which is a very different event.

Re: Feynman on Fermat's Last Theorem

#84

Post by tgb » Sat, Jul 02, 2016, 1:59 PM UTC

Houshalter wrote: post #81

erikbern wrote: post #69

There'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 wr…

What's interesting about this argument is, if its true, it's an example of a simple conjecture that is true but unprovable. But not like Godel sentences, which can be proved under different axioms, or proved to be unprovable. This would be completely unprovable. And not because of some logical paradox, but simply because there is no mathematical reason for it to be false, it's just a coincidence. There are also proba…

Well, any statement can be proven under different axioms. Just take that statement itself as your only axiom.

Re: Feynman on Fermat's Last Theorem

#85

Post by k__ » Sat, Jul 02, 2016, 2:02 PM UTC

Chinjut wrote: post #70

jsprogrammer wrote: post #66

Earlier quoted context omitted.

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?

I think this was a joke

Re: Feynman on Fermat's Last Theorem

#86

Post by tgb » Sat, Jul 02, 2016, 2:04 PM UTC

jordigh wrote: post #83

tgb wrote: post #80

Earlier quoted context omitted.

You're forgetting to integrate the probability over the domain 0 to n. If you do, you always get 1 for n at least 42.

Those are two separate events. I'm talking about the event n = 42, which makes sense for all [n]. You're talking about the event n >= 42, which is a very different event.

No, I'm not talking about integrating the probability that n >= 42.

What's the chance that a random number in [n] is 42? It's 1/n, if n is at least 42. Sum that over all x in [n] and you get 1, i.e. the probability that there is a number in [n] that is 42.

Re: Feynman on Fermat's Last Theorem

#87

Post by utopcell » Sat, Jul 02, 2016, 3:50 PM UTC

By this probabilistic reasoning, we shouldn't bother searching for "meaningful" polynomial time algorithms because they are practically non-existent in the sea of all exponential time algorithms. If anything, this argument should be a motivator to work on a problem because if an underlying structure exists, it would be rare and beautiful. Very nicely written article, showing us a glimpse of the inner workings of a great mind.

Re: Feynman on Fermat's Last Theorem

#88

Post by xiaoma » Sat, Jul 02, 2016, 4:35 PM UTC

lb1lf wrote: post #36

plg wrote: post #23

i have to say this never-ending fixation / worship of Feynman is a bit creepy

There's hardly any denying that Feynman was quite a character; he (sometimes spectacularly) failed to live up to our expectations as to how such a brilliant intellect ought to behave. That is probably a significant reason for his enduring appeal; a streetwise, bongo-playing physicist with a talent for quips.

>"he (sometimes spectacularly) failed to live up to our expectations as to how such a brilliant intellect ought to behave."

Can you elaborate on this?

Re: Feynman on Fermat's Last Theorem

#89

Post by Chinjut » Sat, Jul 02, 2016, 5:45 PM UTC

fareesh wrote: post #78

My math is passably college level, so most of these proofs go over my head. I've always thought of FLT as stating "You cannot make n-3 cubes of side z by stacking/tiling n-3 cubes of side x and y", at which point it becomes something of a tessellation problem in 3 dimensions, which would seem to me to be the intuitive way to go about proving this.

Sorry, I'm having difficulty following you. What do you mean by an "n-3 cube"? Or are you actually talking about n - 3 many different cubes?

Re: Feynman on Fermat's Last Theorem

#90

Post by jsprogrammer » Sat, Jul 02, 2016, 6:12 PM UTC

k__ wrote: post #85

Chinjut wrote: post #70

Earlier quoted context omitted.

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?

I think this was a joke

Yes, we are talking about an anecdote from the book, "Surely You're Joking, Mr. Feynman!": Adventures of a Curious Character

?

Why would Feynman not have published a full(er) proof?