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

TL;DR

This paper introduces a method to determine the solvability of the Diophantine equation x^2 - Dy^2 = n for specific forms of D involving primes with particular congruence properties and quadratic representations.

Contribution

It provides a novel approach for assessing solvability of the equation for D as a product of primes with special congruence and quadratic sum conditions.

Findings

Method applicable to D=pq with specified prime conditions

Method applicable to D=2p_1p_2...p_m with primes ≡ 1 mod 8

Criteria for solvability based on prime congruences and quadratic representations

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 $r,s \equiv \pm 3 \mod 8$.