  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.
