Practical solution of some families of quartic and sextic diophantine hyperelliptic equations

TL;DR

This paper provides elementary number theory methods to solve specific families of quartic and sextic hyperelliptic Diophantine equations over integers, expressing solutions via divisors of discriminants.

Contribution

It introduces a practical approach to find integer solutions of certain hyperelliptic equations using discriminant divisors, expanding solution techniques for these equations.

Findings

Solutions expressed through divisors of the discriminant of f(x)

Applicable to specific quartic and sextic hyperelliptic equations

Provides elementary number theory-based methods

Abstract

Using elementary number theory we study Diophantine equations over the rational integers of the following form, $y^2=(x+a)(x+a+k)(x+b)(x+b+k)$, $y^2=c^2x^4+ax^2+b$ and $y^2=(x^2-1)(x^2-\alpha^2)(x^2-(\alpha+1)^2).$ We express their integer solutions by means of the divisors of the discriminant of $f(x),$ where $y^2=f(x)$.