On the equation $x^2-Dy^2=n$

Abstract

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 $p,q\equiv 1 \mod 4$ are distinct primes with $(\frac{q}{p})=1$ and $(\frac{p}{q})_4(\frac{q}{p})_4=-1$. $(2)$ $D=2p_1p_2... p_m$, where $p_i\equiv 1 \mod 8,1\leq i\leq m$ are distinct primes and $D=r^2+s^2$ with $r,s \equiv \pm 3 \mod 8$.