On the diophantine equation x 2 ? Dy 2 = n

We propose a method to determine the solvability of the diophantine equation x ² ? Dy ² = n for the following two cases:

1. D = pq, where p, q ≡ 1 mod 4 are distinct primes with $(\tfrac{q} {p}) = 1 $ and $(\tfrac{p} {q})_4 (\tfrac{q} {p})_4 = - 1 $
2. D = 2 _p1p2…p _m , where p _i ≡ 1 mod 8, 1 ? i ? m are distinct primes and D = r ²+s ² with r, s ≡ ±3mod8.