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Feynman on Fermat's Last Theorem (2016)

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

#21

Post by Retric »

jordigh wrote: post #13
foxes wrote: post #9

Earlier quoted context omitted.

This isn't a proof.

I know, but it's being exhibited as an example of how back-of-the-envelope type of approximations by physicists can be just as good as rigid mathematical thinking. And I don't find this to be a convincing example of how loose physicist arguments can work. Schwartz distributions, infintesimals; okay, fine, those turned out to be a weird trick that can be formalised. But sometimes their tricks are just plain wrong and…

That’s not what’s happening.

Consider, many useful primality tests are statistical in nature. It’s pure math, and exact answer is possible but it’s still useful to get a quick check to see if something is a waste of time.

Really, if a full solution takes 20 years you don’t want to actually spend 20 years without having a very good idea it’s going to work.

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

#22

Post by ColinWright »

ox_n wrote: post #2

_DON'T DOWN VOTE JUST BECAUSE YOU CAN'T DO MATH_ The proof Fermat hinted to was about the difference between squares. All whole numbers taken to a power greater than two (n^3) can be represented as the difference between two whole squares (x^2 - y^2). These differences can then be shown as the sum of consecutive odd numbers: 2^3 = 3^2 - 1^2 = (1+3+5) - (1) = 8, 3^3 = 6^2 - 3^2 = (1+3+5+7+9+11) - (1+3+5) = 27, 4^3 = 1…

I'm working to understand this, but I can't seem to fit it together. Following Feynman's lead in this sort of thing, can you give me an explicit example of why the equation x^13+y^13=z^13 has no solutions? Or even just use your technique to explain why x^5+y^5=z^5 has no solutions?

Thanks.

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

#23

Post by dang »

ox_n wrote: post #2

_DON'T DOWN VOTE JUST BECAUSE YOU CAN'T DO MATH_ The proof Fermat hinted to was about the difference between squares. All whole numbers taken to a power greater than two (n^3) can be represented as the difference between two whole squares (x^2 - y^2). These differences can then be shown as the sum of consecutive odd numbers: 2^3 = 3^2 - 1^2 = (1+3+5) - (1) = 8, 3^3 = 6^2 - 3^2 = (1+3+5+7+9+11) - (1+3+5) = 27, 4^3 = 1…

Can you please read the site guidelines and follow them?

https://news.ycombinator.com/newsguidelines.html

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

#24

Post by x3tm »

ox_n wrote: post #17
x3tm wrote: post #4

Earlier quoted context omitted.

> It's not that hard people. Stop believing everything you're told about how "hard" something is. There are still many problems in physics and mathematics which are considered "hard" (e.g., dark energy, Riemann hypothesis, etc). Can we crack them by simply adopting your positive mindset?

What other mindset do you see working better?

I don't think the "you can do anything" mindset works in real life. It helps self-help book authors sell their stuff, but it's not a good strategy to live by. (Incidentally, this reminds me of Key & Peele's "You can fly" sketch).

What does work though is this: advanced formal education in a topic. Once you have that you can start thinking on how to solve some simple open problems. And if you are lucky and turn out to be extremely smart, you may be able to tackle more challenging problems. Some amount of self confidence may also you to keep going but doesn't make you a genius overnight.

Simply going to a mindset where things are 'not hard' is closer to delusion than it is to anything else.

In academia we get often emails from people who solved quantum gravity (e.g. using fire), show us how einstein is wrong (e.g. using a pendelum), etc. I'm pretty sure they also convinced themselves to "Stop believing everything they're told about how "hard" something is"

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

#25

Post by jordigh »

fanzhang wrote: post #16
jordigh wrote: post #5

This proof (or "plausibility argument") bugs me so much. Just because something thins out and becomes rare doesn't mean it doesn't exist. As n gets bigger, the probability of n being a perfect square gets smaller and smaller. In the limit, the probability is zero. Does this mean square numbers don't exist?

By Feynman's argument, you can prove that square numbers almost certainly keep on existing. Roughly, it goes as such: 1) the probability of N being a perfect square is proportional to 1/sqrt(N). 2) For any N_0 arbitrarily high, if you integrate from N_0 to infinity the expression (1/sqrt(N) dN), you get infinity. 3) The expression in 2) is the "Feynman equivalent" of the expected number of square numbers above N_0. S…

Okay, let's pick something rarer. Rational numbers.

If you integrate the characteristic function of the rational numbers over any interval, you get zero because rational numbers are very rare.

So they don't exist either?

To be less glib, I don't see Feynmann's argument to be bringing anything new. We already knew that counterexamples, if they existed, would be very rare because we tried looking for them with computers and we couldn't find them. But stuff being rare still doesn't prove anything.

Many of us were fooled by Skewe's number:

https://en.wikipedia.org/wiki/Skewes%27s_number

There's no way to conclude that this exists via brute calculation. It's just inconceivably large and would have eluded any of Feynmann's methods.

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

#27

Post by andrepd »

ox_n wrote: post #15
ColinWright wrote: post #10

Earlier quoted context omitted.

Do you really believe that: (a) This constitutes a proof; (b) This is the "proof" that Fermat had; (c) Mathematicians missed this for over 350 year? I'm not quite sure exactly what you are claiming.

It's completely arrogant to assume that because it hasn't been solved by "better" people that I couldn't solve it.

>about: Fuck you, hater.

Oh, you're that guy.

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

#28

Post by VRay »

x3tm wrote: post #24
ox_n wrote: post #17

Earlier quoted context omitted.

What other mindset do you see working better?

I don't think the "you can do anything" mindset works in real life. It helps self-help book authors sell their stuff, but it's not a good strategy to live by. (Incidentally, this reminds me of Key & Peele's "You can fly" sketch). What does work though is this: advanced formal education in a topic. Once you have that you can start thinking on how to solve some simple open problems. And if you are lucky and turn out to…

Oh man, that reminds me of an experience I had in college. I was working with the aerospace department on their fusion reactor (I was just writing software to help them process data from it, not involved in the science itself). My boss kept getting calls from crackpots who'd go on and on and on about their bogus theories, and how they were being shut out of the mainstream by small minded fools, etc etc.

It was pretty frustrating. He was too nice a guy to tell them off or even cut them off quickly.

My advice to any crackpots who are really sure they're actually geniuses: Get into the stock market (with a SMALL investment). If you're as smart as you think you are, you can find an angle and turn $100 into $1,000,000 or more, and then if anything it'll be GOOD that nobody ever believed in you. I've run across arbitrage opportunities that would have made me fiendishly rich if I'd noticed them sooner myself, believe it or not. Just be careful and don't mess with box spreads.

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

#29

Post by andrepd »

ox_n wrote: post #19
yters wrote: post #7

Earlier quoted context omitted.

Sounds interesting, but not sure what you mean by: > you'll discover that there will always be a gap if you try and combine two odd number series together Can you elaborate?

Consecutive base numbers will necessarily alternate between even and odd. So even the closest base numbers still have a gap between their resulting odd number series, which only increases as the distance between base numbers increases.

Still don't get it. What do you mean by "base numbers" and what do you mean by "alternate between even and odd"?

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

#30

Post by nestorD »

syn0byte wrote: post #26

Is anyone still trying to come up with Fermat's original "truly marvelous proof"? Or have math folk talked themselves out of its possible existence?

I recall reading that there is an, incorrect, proof that would match the kind of proof we expect from Fermat and thus is believed to be the one he had in mind. However I was unable to find it. Instead I found a discussion on how likely it is that he had a proof : https://hsm.stackexchange.com/questions/3/what-evidence-is-t...
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