There is no Diophantine quintuple

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 $1 \le i < j \le m$. 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.