Pells equation

The square root of two is often approximated 1.414..., which some might interpret as 1.41414141414..., or 140/99. The reciprocal? of the [square root of two]? to three [decimal place]?s is 0.707, which is suggestive of 0.70707070..., or 70/99. If 70/99 approximates the reciprocal of the square root of two, it follows that 99/70 approximates the square root of two. As it turns out, the square root of two is between 140/99 and 99/70. The arithmetic mean of these two rationals is 19601/13860. That number squared is 384199201/192099600. It turns out that 2 times the denominator? 192099600 is 384199200, which differs from the numerator? by only one. p=19601 and q=13860 satisfies the [Diophantine equation]? 2q²+1=p². Any combination of p and q that satisfy this equation will be a reasonably good approximation for the square root of two. More generally, a combination of p and q that satisfies nq²+1=p² (Pell's equation) is a reasonably good approximation for the square root of n. It turns out that if (p,q) satisfies Pell's equation, so does (2pq,2q²-1). It also turns out that p and q can be found to satisfy Pell's equation for any natural number n that is not a [perfect square]?. Given a computer with bignum capability, this makes it easy to converge rapidly toward any irrational square root of a positive integer. As an added bonus, Pell's equation can always be solved in a finite number of steps by calculating the [continued fraction]? representation of an irrational square root of a natural number.