yield solutions to Pell's equation. This can be understood in part to arise from the fact that successive convergents of a continued fraction share the same property: If pk−1/qk−1 and pk/qk are two successive convergents of a continued fraction, then the matrix has determinant (−1)k. Smooth numbers Størmer's theorem applies Pell equations to find pairs of consecutive smooth numbers, positive integers whose prime factors are all smaller than a given value. As part of this theory, Størmer also investigated divisibility relations among solutions to Pell's equation; in particular, he showed that each solution other than the fundamental solution has a prime factor that does not divide n. The negative Pell's equation The negative Pell's equation is given by and has also been extensively studied. It can be solved by the same method of continued fractions and has solutions if and only if the period of the continued fraction has odd length. However, it is not known which roots have odd period lengths, and therefore not known when the negative Pell equation is solvable. A necessary (but not sufficient) condition for solvability is that n is not divisible by 4 or by a prime of form 4k + 3. Thus, for example, x2 − 3ny2 = −1 is never solvable, but x2 − 5ny2 = −1 may be. The first few numbers n for which x2 − ny2 = −1 is solvable are 1, 2, 5, 10, 13, 17, 26, 29, 37, 41, 50, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97, ... . Let . The proportion of square-free n divisible by k primes of the form 4m + 1 for which the negative Pell's equation is solvable is at least α. When the number of prime divisors is not fixed, the proportion is given by 1 - α. If the negative Pell's equation does have a solution for a particular n, its fundamental solution leads to the fundamental one for the positive case by squaring both sides of the defining equation: implies As stated above, if the negative Pell's equation is solvable, a solution can be found using the method of continued fractions as in the positive Pell's equation. The recursion relation works slightly differently however. Since , the next solution is determined in terms of whenever there is a match, that is, when is odd. The resulting recursion relation is (modulo a minus sign, which is