On a conjecture Mordell

TL;DR

This paper proves Mordell's conjecture for four specific infinite families of primes congruent to 3 mod 4, extending the verified cases beyond previous computational limits.

Contribution

It establishes the validity of Mordell's conjecture for four new infinite prime families, advancing understanding of the conjecture's scope.

Findings

Mordell's conjecture holds for four infinite prime families.

The conjecture verified for primes up to 10^7.

New theoretical proof extends previous computational verification.

Abstract

A conjecture of Mordell states that if $p$ is a prime and $p$ is congruent to $3$ mod $4$, 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.