Pell's Equation - The Negative Pell Equation

The Negative Pell Equation

The negative Pell equation is given by

(eq.1)

It has also been extensively studied; it can be solved by the same method of using continued fractions and will have solutions when the period of the continued fraction has odd length. However we do not know which roots have odd period lengths so we do not know when the negative Pell equation is solvable. But we can eliminate certain n since a necessary but not sufficient condition for solvability is that n is not divisible by a prime of form 4m+3. Thus, for example, x2-3py2 = -1 is never solvable, but x2-5py2 = -1 may be, such as when p = 1 or 13, though not when p = 41.

Cremona & Odoni (1989) demonstrate that the proportion of square-free n divisible by k primes of the form 4m+1 for which the negative Pell equation is soluble is at least 40%. If it does have a solution, then it can be shown that its fundamental solution leads to the fundamental one for the positive case by squaring both sides of eq. 1,

to get,

Or, since ny2 = x2+1 from eq.1, then,

showing that fundamental solutions to the positive case are bigger than those for the negative case.