Abstracted one generator for approximate square root of 3

Found myself intrigued by putting up just enough information to do the calculations for square root of 3 with one of an infinity of possible relations:

sqrt(3) approximately equals x_n+1/y_n+1, where:

x_n+1 = 97x_n + 168y_n

y_n+1 = 56x_n + 97y_n

and x₀ = 1, and y₀ = 0;

Does an n exist for: x = 1351, and y = 780?

(If you figure it out, and answer is yes, please give n in comments.)

With those if it does then: x_n+1 = 262087, and y_n+1 = 151316

And 262087/151316 equals approximately: 1.73205080758

Correct for sqrt(3), same number of digits: 1.73205080756

And differs from sqrt(3) by approximately: 1.26e-11

That pc calculator is so handy.

Oh yeah, negative DOES work. Just been kind of sloppy, and just talking the positive, so you can start with: x₀ = -1, and y₀ = 0

The math does NOT care. And of course you have two solutions! Positive and negative for sqrt(3). So yes, try x and y with different negations and see what happens! (Still works.)

Wondering if should fix prior posts, occuring to me don't really care as is freaking obvious anyway. Moving on to other more important things.

Still just doing these for fun.


James Harris