On a conjecture of Mordell

Abstract

A conjecture of Mordell states that if $p$ is a prime and $p$ is congruent to $3$ modulo $4$, then $p$ does not divide $y$ where $(x,y)$ is the fundamental solution to $x^{2}-py^{2}=1$. The conjecture has been verified for primes not exceeding $10^{7}$. In this article, we show that Mordell's conjecture holds for four conjecturally infinite families of primes.