### 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!

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:
32+42=52;
202+212=292;
1192+1202=1692;
.
.
.

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. :)

1. chilayse on

So…you noticed a pattern in math that no one else found before? *so very lost at the numbers* that’s neat. ^_^

• The Rev Dr Sherwood Forrester on

Pretty much, yeah. I really didn’t think there was anything original left to find among the set of solutions to Pythagoras… it’s meaningless, but cool. Especially when you think this means I caught something Fermat missed. :)

• chilayse on

The best I can do is poke at my friends sudoku puzzles she gets stuck on and solve them in minutes when she spent days on them…right after she told me the rules.

I’m ok with numbers but I have very little interest or education with them so *shrugs* My school was so bad in math that when one of my friends went to the university and said she wanted to be a computer science major they said ok you’ll need remedial math….simply because of our highschool.

Ah I see the edit….but at least you figured it out all by yourself? ^^? That’s something special too right?

2. argh_jim on

Fermat DID find it, and he had a simple solution, but it got lost in the mail….

3. nsingman on

As a math geek who did some work in high school on Pythagorean triple generators, let me just say “Cool!” :-)

• The Rev Dr Sherwood Forrester on

I came to it, or perhaps more accurately came back to it, after finally discarding Fermat, having become convinced that he did not have a complete proof but had misgeneralized a limited-case proof. I’m not sure that a ‘classical’ proof of Fermat exists–Wiles’ monumental task requires 20th century techniques, and invented what will probably be a large chunk of 21st century math (to say nothing of at a stroke validating every otherwise valid proof that contained ‘Assuming Taniyama-Shimura…’).

The lovely thing about Pythagoras is that it’s so rich, so full of patterns. It’s like rooting around in π: there’s always room to find something no one’s seen before, or even accidentally rediscover the work of the professionals.

I’m actually not disappointed to see that someone else did notice it, because I expected that. I’m just disappointed that I didn’t come forward with my finding earlier, because I’ve had this number tucked away for a few years, and the date of the other finding was only two years ago.

On the plus side, the latest new finding regarding just this particular sequence was added a week ago today, so like I said, there’s always room for more. :)