You mean no one’s noticed in the last 2500 years?
EDIT: Well, yes, someone *has* noticed, and a slightly more thorough search of the (online) literature shows that my discovery had already been discovered; I’ve emailed the professor to let him know I’ve found it. Science demands no less, of course! Heck, I don’t mind independently rediscovering things, at least it means I’m doing something right… but it was fun while it lasted!
This is where I lose just about everyone, but bear with me, I’m kinda excited about this in an übergeeky way.
I love numbers. I do math puzzles for fun. I dallied with Fermat (for all n>2 there are no integer solutions for an+bn=cn) for a while, proving for my own satisfaction the cases for n=3, n=4 and n=5, even after Wiles ended the game once and for all. I independently (re)discovered Newton’s method for extracting roots.
Mainly, I’ve always had an odd fascination with integer solutions to the Pythagorean Theorem, to wit: a2+b2=c2
Anyway, the family of solutions I got interested in are of the form b = a + 1, as in:
I happened to notice a fairly simple ratio between the successive solutions, and forgot mostly about it until this morning when I happened to be talking about math geekery with a friend online. He mentioned Dr. Ron Knott’s Pythagorean Triples site, and I looked into the section on consecutive legs (which is the part I was playing with — a2+(a+1)2=c2 — and much to my surprise, there’s no mention of the ratio, although there are several methods for constructing numbers that will solve the equation. Hmm! says I to myself, and email my findings off to Dr. Knott, fully expecting an email in a week or so saying that yes, this ratio was first noticed by (so and so) in (seventeen fifty-something) or some other thing like that.
I got a response in a little more than an hour that my discovery appears to be new.
He offered to submit it to the Online Encyclopedia of Interesting Sequences under my name with his comments (he makes it sound more like math than I do), and I asked him to please go right ahead because I haven’t the faintest idea how to so do.
I guess that says something about The Theorem, if it still has secrets to reveal two and a half millennia after Pythagoras … I have no idea what it says about me, other than that I have a certain gift for numbers, or I probably have too much time on my hands… or both. :)