Diophantine quintuple conjecture

Importance: Medium ✭✭
Subject: Number Theory
Recomm. for undergrads: yes
Posted by: maxal
on: November 22nd, 2008

\begin{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 $1 \leq i < j \leq m$. \end{definition}

\begin{conjecture}[1] Diophantine quintuple does not exist. \end{conjecture}

It would follow from the following stronger conjecture \cite{Da}:

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

It was proved in \cite{Db} that there are only finitely many Diophantine quintuples and no Diophantine sextuples.

Conjecture (2) is motivated by an observation of \cite{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 \href[Diophantine $m$-tuples]{http://web.math.hr/~duje/dtuples.html}, a survey of the main problems and results concerning Diophantine m-tuples.

[Db] A. Dujella, \href[There are only finitely many Diophantine quintuples]{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=}, 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.