Diophantine quintuples containing triples of the first kind

TL;DR

This paper proves that no Diophantine quintuples include triples of the first kind, which are specific triples with a large element relative to the second.

Contribution

It establishes a non-existence result for Diophantine quintuples containing triples of the first kind, advancing understanding of their structure.

Findings

No Diophantine quintuples contain triples of the first kind.

The result constrains the possible configurations of Diophantine quintuples.

The proof involves analyzing the properties of these special triples.

Abstract

We consider Diophantine quintuples $\{a, b, c, d, e\}$, sets of distinct positive integers the product of any two elements of which is one less than a perfect square. Triples of the first kind are the subsets $\{a, b, d\}$ with $d> b^{5}$. We show that there are no Diophantine quintuples containing triples of the first kind.