Searching for Diophantine quintuples

TL;DR

This paper investigates Diophantine quintuples, sets of five positive integers with pairwise products one less than a perfect square, and significantly tightens the upper bound on their possible number, supporting the conjecture of their non-existence.

Contribution

The authors improve the upper bound on the number of Diophantine quintuples from previous estimates to at most 1.18×10^{27}, advancing understanding of their rarity.

Findings

Upper bound on Diophantine quintuples is now 1.18×10^{27}.

Supports the conjecture that no such quintuples exist.

Provides refined estimates for the distribution of these sets.

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 $1.18\cdot 10^{27}$ Diophantine quintuples.