That was very cool-- a CAPTCHA that uses number theory to calculate Pi as a side effect.....
Good reading at that link. Thanks!
One of the oddest things about math, to most people, is how a constant like pi, from geometry, appears in all kinds of things that have little or nothing to do with circles.
I still remember having to figure this out for a takehome exam in number theory.
This is a poorly-formed question, because there's no such thing as a uniform randomly-chosen integer. It's more accurate to say that the probability of gcd(m, n) = 1, with m and n chosen from {1, ..., N}, approaches 6/pi^2 as N -> infinity.