I would have preferred to do this sober, but as some may know, I'm a bit under the weather (chronic fatigue) so it takes a stimulant, to get me on line; given the local quacks' refusal to take this malady seriously, my only recourse is alcohol, so please excuse me if the following is not totally coherent?
Back in June, when I was on a (distressingly) short high cycle, I read a book by Simon sings No, Singh! And why not; the Indian contribution to mathematics is enormous. That said, there were some curious accounts in the book that made no sense to me, like Riemann's hypothesis, about needing complex numbers to understand the distribution of Primes (but what can you expect from some guy who wastes his time on non-Euclidean geometry...BUGGER, just remembered that Gauss did the same, and he was even smarter than me!).
Another klutz was this guy Fermat with his method of infinite division or some such (which I couldn't understand, and so cannot remember) but there's the give-away; the book was about Fermat's last theorem, supposedly the most difficult problem of all time, which had defeated the worlds greatest mathematicians for three and a half centuries (but let's face it, these mathematicians are weird, right?). Admittedly, it was a tough problem, and without the material in Singh's book, I could not have solved it (and even then it took me a full two weeks!).
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 problem?
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?
06 Aug 2007, 3:29 pm
