Integers representable as a difference of two rational fourth powers

TL;DR

This paper classifies all positive integers up to 10,000 that can be expressed as the difference of two rational fourth powers, extending previous work on sums of fourth powers.

Contribution

It provides a complete list of positive integers up to 10,000 representable as a difference of two rational fourth powers, filling a gap in number theory research.

Findings

Identifies all such integers up to 10,000

Completes the classification for the difference of two rational fourth powers

Builds on previous work on sums of fourth powers

Abstract

In Section 6.6 of the book {\it Number Theory, Volume I: Tools and Diophantine Equations, Graduate Texts in Mathematics, Volume 239, Springer (2007)}, Cohen investigated the solubility of the equation $n=x^4+y^4$ in the rational numbers $x,y$ for all positive integers $n \leq 10000$. Motivated by this, we investigate the equation $n=x^4-y^4$ and obtain the complete list of positive integers $n\leq 10000$ that can be represented in this form for some nonzero rational numbers $x$ and $y$.