Bounds on the number of Diophantine quintuples

TL;DR

This paper establishes an improved upper bound of approximately 1.9×10^{29} on the number of Diophantine quintuples, sets of five positive integers with pairwise products one less than a perfect square.

Contribution

The authors provide a significantly tighter upper bound on the number of Diophantine quintuples, advancing the understanding of their possible existence.

Findings

Upper bound of 1.9×10^{29} on Diophantine quintuples

Supports the conjecture that no such quintuples exist

Refines previous estimates significantly

Abstract

We consider Diophantine quintuples $\{a, b, c, d, e\}$. These are sets of distinct positive integers, the product of any two elements of which is one less than a perfect square. It is conjectured that there are no Diophantine quintuples; we improve on current estimates to show that there are at most $1.9\cdot 10^{29}$ Diophantine quintuples.