There is no Diophantine quintuple

Abstract

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.