[ox_n]

_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 = 10^2 - 6^2 = (1+3+5+7+9+11+13+15+17+19) - (1+3+5+7+9+11) = 64

    5^3 = 15^2 - 10^2 = (21+23+25+27+29) = 125

When you examine the odd number series that results from each base, you'll discover that there will always be a gap if you try and combine two odd number series together, which explains Fermat's little joke about margins. The same trick works for higher powers.

It's not that hard people. Stop believing everything you're told about how "hard" something is.

HINT: The number of odd numbers in the series exactly matches the starting square base number

[dang]

Can you please read the site guidelines and follow them?

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

[ColinWright]

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.

[ColinWright]

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.

[throwawayznxnyn]

This argument puzzled me, but unfortunately this is as far as I got:

~~Lemma 1~~:

    z^n can be written as a difference of squares:

Proof:

z^n = x^2 - y^2 = (x+y)(x-y), leading to the system of equations

x + y = z^(n-1)

x - y = z

which may be solved for x and y:

x = (z^(n-1) + z)/2

y = x - z

QED.

~~Lemma 2~~:

    Any number squared can be written as a sum of sequantial odd numbers starting at 1.

By induction:

(n)^2 = (n-1)^2 + (2(n-1)+1) = \sum_{k=0}^{n-1}(2k + 1)

QED.

~~~Fermat's Last Theorem~~~:

    z^n = x^n + y^n has no solutions for n>2, and positive z, x, y.

Proof:

Without loss of generality, assume a > b, then rewrite z^n as a sum of sequential odd numbers starting at b:

z^n = a^2 - b^2 = \sum_{k=b}^{a-1}(2k+1)

x^n and y^n can similarly be written as a sum of sequenatal odd numbers:

x^n = c^2 - d^2 = \sum_{k=d}^{c-1}(2k+1)

y^n = e^2 - f^2 = \sum{k=f}^{e-1}(2k+1)

By substitution to the Theorem's equation:

\sum_{k=b}^{a-1}(2k+1) = \sum_{k=d}^{c-1}(2k+1) + \sum_{f}^{e-1}(2k+1)

Which is true if and only if there are no gaps in the bounds of summation on the right side, so d = b, c = f, and e = a. But then

a^2 - b^2 = (f^2 + (-a^2))^n + (b^2 + (-f^2))^n

and this is certainly not true by the binomial theorem. We've reached a contradiction so QED.

^^ This is where I'm stuck. I don't actually know why that would not be true by the binomial theorem? It seems like simple expansion should do it.

[HNLurker2]

Off topic: why doesn't HN support LaTeX?

[nappy-doo]

Likely because MathML isn't widely supported, and has even been removed from Chrome.

[ColinWright]

MathJax works an absolute treat.

[gowld]

Except on Android where the math takes up more horizontal space than the layout engine thinks it should, causing overlap with text after the math.

[ColinWright]

I'll pass that report on to them - have you reported it already? In which browser is that happening?

[yters]

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?

[aaaaaardvark]

ox_n's last theorem

[ox_n]

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.

[andrepd]

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

[gowld]

'ox_n appears to be trying to prove (by a pigeonhole agument) that the equation x^n + y^n = z^n in unsolvable for _some z_. That's much weaker than proving that it is unsatisfiable at _every_ z.

[x3tm]

> 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?

[ox_n]

What other mindset do you see working better?

[x3tm]

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"

[VRay]

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.

[syn0byte]

So what your saying is... We should ignore letters from patent clerks?

[ColinWright]

No, but even those from "left field", if genuine, tend to take the time and care to write things up properly, to use the nomenclature of the field, to address obvious potential concerns up front.

If you're asserting something that's likely to encounter resistance, it's worth being clear and careful.

[keymone]

> The same trick works for higher powers.

can you demonstrate?

[ox_n]

Yes.

x and y will be a multiple of the base number.

[keymone]

i don't get it, how do you go from difference of squares turning into sequence of odd numbers to absence of power of some integer?