A set of
m
m
positive integers
{
a
1
,
a
2
,
…
,
a
m
}
\{a_1, a_2, \ldots , a_m\}
is called a Diophantine
m
m
-tuple if
a
i
a
j
+
1
a_i a_j + 1
is a perfect square for all
1
≤
i
>
j
≤
m
1 \le i > j \le m
. 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
m
-tuples states that no Diophantine quintuple exists at all. We prove this conjecture.