Pell's equations \(X^ 2-mY^ 2=-1,-4\) and continued fractions (Q1078225)

Let \(m\equiv 1(mod 4)\) be a positive nonsquare integer such that the equation \(x^ 2-my^ 2=-1\) is solvable in integers x,y. The authors show that the equation \(x^ 2-my^ 2=-4\) is solvable in coprime integers x,y if and only if \(\ell (\sqrt{m})\equiv \ell ((1+\sqrt{m}))(mod 4),\) where \(\ell (\alpha)\) denotes the length of the period of the continued fraction expansion of the quadratic irrational \(\alpha\). Further congruences modulo 4 for \(\ell (\sqrt{m})\) and \(\ell ((1+\sqrt{m}))\) are given when \(m\equiv 5(mod 8)\).