paper

There is no Diophantine quintuple

arXiv:1610.04020

Abstract

A set of $m$ positive integers ${a_{1}, a_{2}, \dots, a_{m}}$ is called a Diophantine $m$ -tuple if $a_{i} a_{j} + 1$ is a perfect square for all $1 \leq i < j \leq m$ . In 2004 Dujella proved that there is no Diophantine sextuple and that there are at most finitely many Diophantine quintuples. In particular, a folklore conjecture concerning Diophantine $m$ -tuples states that no Diophantine quintuple exists at all. In this paper we prove this conjecture.