# Diophantine quintuple conjecture

**Definition**A set of m positive integers is called a Diophantine -tuple if is a perfect square for all .

**Conjecture (1)**Diophantine quintuple does not exist.

It would follow from the following stronger conjecture [Da]:

**Conjecture (2)**If is a Diophantine quadruple and , then

It was proved in [Db] that there are only finitely many Diophantine quintuples and no Diophantine sextuples.

Conjecture (2) is motivated by an observation of [AHS] that every Diophantine triple can be extended to a Diophantine quadruple

## Bibliography

[Da] A. Dujella Diophantine -tuples, a survey of the main problems and results concerning Diophantine m-tuples.

[Db] A. Dujella, There are only finitely many Diophantine quintuples, J. Reine Angew. Math. 566 (2004), 183-214.

[AHS] J. Arkin, V. E. Hoggatt and E. G. Strauss, On Euler's solution of a problem of Diophantus, Fibonacci Quart. 17 (1979), 333-339.

* indicates original appearance(s) of problem.

## This result has been proven

in a paper announced in 2016 and published in 2019, He, Togbé and Ziegler [350] gave the proof of the Diophantine quintuple conjecture