Diophantine quintuple conjecture

Importance: Medium ✭✭
Subject: Number Theory
Recomm. for undergrads: yes
Posted by: maxal
on: November 22nd, 2008
Definition   A set of m positive integers $ \{a_1, a_2, \dots, a_m\} $ is called a Diophantine $ m $-tuple if $ a_i\cdot a_j + 1 $ is a perfect square for all $ 1 \leq i < j \leq m $.
Conjecture  (1)   Diophantine quintuple does not exist.

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

Conjecture  (2)   If $ \{a, b, c, d\} $ is a Diophantine quadruple and $ d > \max \{a, b, c\} $, then $ d = a + b + c + 2bc + 2\sqrt{(ab+1)(ac+1)(bc+1)}. $

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 $ \{a,b,c\} $ can be extended to a Diophantine quadruple $ \{a,b,c,a + b + c + 2bc + 2\sqrt{(ab+1)(ac+1)(bc+1)}\}. $


[Da] A. Dujella Diophantine $ m $-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

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