Prime Numbers

2.3 Squares and roots 107
t = p − b 2 ;
if(t ≡ 0(modd)) return {}; // Return empty.
if(t/d not a square) return {}; // Return empty.
return (±b, ± t/d); // Solution(s) found.
This completely solves the computational Diophantine problem at hand. Note
that an integer square-root finding routine (Algorithm 9.2.11) is invoked at
two junctures. The second invocation—the determination as to whether t/d is
a perfect square—can be done along the lines discussed in the text following
the Algorithm 9.2.11 description. Incidentally, the proof that Algorithm 2.3.12
works is, in words from [Cohen 2000], “a little painful.” There is an elegant
argument, due to H. Lenstra, in [Schoof 1995], and a clear explanation from
an algorist’s point of view (for d = 1) in [Bressoud and Wagon 2000, p. 283].
The second case, namely for the Diophantine equation x 2 + |D|y 2 =4p,
for D