Pell's equation

Pell's equation is an equation whose solutions yield good rational approximations to square roots of natural numbers.

As motivation, consider the square root of two. It is often approximated 1.414..., which some might incorrectly interpret as 1.41414141414..., or 140/99. Likewise, the reciprocal of the square root of two[?] to three decimal places[?] 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 fraction of natural numbers p and q that satisfy this equation will be a reasonably good approximation for the square root of two.

More generally, if n is a given natural number, then any fraction of natural numbers p and q that satisfy Pell's equation

nq² + 1 = p²

is a reasonably good approximation for the square root of n. The larger the numbers p and q, the better the approximation. It turns out that if (p, q) satisfies Pell's equation, then so does (2pq, 2q²-1). It also turns out that p and q can always 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 n. As an added bonus, Pell's equation can always be solved in a finite number of steps by calculating the continued fraction representation of the square root of n.