sqrt(3) approximately equals

**x**, where:

_{n+1}/y_{n+1}**x**

_{n+1}= 362x_{n}+ 627y_{n}**y**

_{n+1}= 209x_{n}+ 362y_{n}and x

_{0}= 1, and y

_{0}= 0;

Which is just for fun, as have noted every time I put up one of these that of course you can just use a calculator. But it is interesting to me, if I just came across that myself, maybe could make me a bit curious. If you try simplest case x

_{1}= 362 and y

_{1}= 209, so 362/209 approximates sqrt(3). And can see how well it works there.

(362/209)

^{2}approximately equals: 3.0000228

Where clipped a bit as no reason to show all of what pc calculator gave.

Iterate once, and you get, x

_{2}= 262087, y

_{2}= 151316

And: (262087/151316)

^{2}approximately equals 3.00000000004367

Where arbitrarily clipped again. And that method goes to infinity, so you can iterate as much as you like.

Will show one more.

Iterate again: x

_{3}= 189750626 and y

_{3}= 109552575

(189750626/109552575)

^{2}approximately equals 3.0000000000000000833

And 189750626/109552575 approximately equals 1.732050807568877 showing only digits that match with sqrt(3).

For a mathematician? Probably a trivial thing. Is easily explained how it works, and goes back to number theory that is considered rather old. Though I used my own tool to find it, so for me that part is cool. Does my approach work further? Nope. What I did works for sqrt(2) and sqrt(3) and nothing else. Though I do wonder if I were more clever if could figure out how to get something from it to work further. Tantalized by the clues.

Am confident that if you program it into a computer you can quickly exceed its capacity to check the result. But still, is just the sqrt(3).

So why bother with it.

Not even a question really. For me? I like playing with numbers. For some maybe there is some mystery, like why does it work? How does it work?

I think that's cool.

Not every one wants to be a math expert and look at only advanced mathematics that pushes the limits of even the greatest minds on the planet to understand.

I know I don't. Sometimes I just like to play with numbers, just for fun. Sharing that? Why not?

So it's easy mathematics. Guess what. I like easy.

James Harris