On Regular Higher Power Rational Diophantine Triples

TL;DR

This paper investigates rational Diophantine triples where the product of any two elements plus one is a fourth power, producing infinite families with positive elements and discussing challenges for higher powers.

Contribution

It introduces new infinite families of rational triples with products plus one as fourth powers and explores extensions to sixth and eighth powers.

Findings

Infinitely many positive rational triples with products plus one as fourth powers.

Methodology extends to some infinite families, but faces difficulties for higher powers.

Brief discussion on challenges in extending to sixth and eighth powers.

Abstract

A rational Diophantine $m$-tuple is a set $\{a_1,\ldots,a_m\}$ of distinct nonzero rational numbers such that $a_i a_j+1$ is a square for all $1\leq i < j\leq m$. Similarly, we may ask when $a_ia_j+1$ is a $k$-th power. Here, we study the case $k=4$ and produce some non-trivial infinite families of such triples. We show that there are infinitely many triples with positive elements for $k=4$. We also briefly consider the $k=6$ (sextic) and $k=8$ (octic) cases, explaining the difficulties in extending the method to higher exponents.