Listen up: this is gwynfryn's first theorem (transcibed)

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THIS HAS BEEN EDITED: (for the preliminaries, check out the introductions section, or my e-mail: [email protected]]

But never mind all that; what I propose is a much simpler issue (but no less interesting I hope?). Shall we call it gwynfryn's first theorem
While studying Fermat's supposedly intractable proposition; that X to the nth power, plus Y to the nth power = Z to the nth power (the old Pythagoras triangle thing) can only be true when n =2, I was needfully involved with Pythagorean triples (those right-angle triangles of whole number sides which followed the rule, of which there are an infinite number, starting from three, four, five) and could not help noting that, in the smallest case, which could not be reduced, the hypotenuse, the longest side was always odd!

I tried to reason this out (naturally) and imagined that, while a Pythagoras triangle of even numbered sides could exist, it was subject to division until at least one side was an odd number. At this point it becomes obvious that at least two sides must be odd (as two odds make an even, but two evens never make an odd) so why not X is odd and Y is odd and Z (the longest side) is even; why is it always the case that only one of the shorter sides is odd, and Z s always odd?

It turns out it must be so, and I have this marvellous proof, which...Oh dear, I'm getting dreadfully tired; still, some of you geniuses should be able to work this out, right?

Last edited by gwynfryn on 15 Aug 2007, 12:50 pm, edited 1 time in total.

You are taking the piss right?

If X and Y are odd, then the sum of their squares is congruent to 2 mod 4.
However, any square of an even number Z is congruent to 0 mod 4. Hence,
X and Y cannot both be odd.

Forgive, me; I thought this was the maths section!

Dear Thoca, I do believe that Fermat, like myself, was quite unaware of moduluses (should that be moduli?) so I can't ascribe to your "solution".

That said, my solution (if I can ever get a response from those mathematicians who could referee my solution) could be written on one page; it involves neither elliptical equations (which I vaguely remember from my engineering degree) nor these mods you quote (which I'd never heard of until I read Singh's book; drunk as I am I could explain the notion...and if I recover my usual health, I may be able to prove what my instinct tells me, OK, not tonight).

I didn't make much sense last time did I!

What I meant was I'd never heard of mods until I read that book about Fermat, so have no idea if your answer is correct or not (so I'll let you thrash it out with better educated mathematicians than I).

The solution I came up with can be written in one paragraph, and requires no more than pre college algebra, and a reasonable grasp of English, so, while perhaps problematical for NTs, it should be well within the grasp of most autistics.