Definition:Diophantine m-Tuple

Definition

A Diophantine $m$-tuple set of $m$ positive integers such that the product of any $2$ of them plus $1$ is a square number.

Particular Instances

Diophantine Triple

A Diophantine triple is a Diophantine $m$-tuple where $m = 3$.

That is, a Diophantine triple is a set of $3$ positive integers such that the product of any $2$ of them plus $1$ is a square number.

Diophantine Quadruple

A Diophantine quadruple is a Diophantine $m$-tuple where $m = 4$.

That is, a Diophantine quadruple is a set of $4$ positive integers such that the product of any $2$ of them plus $1$ is a square number.

Diophantine Quintuple

A Diophantine quintuple is a Diophantine $m$-tuple where $m = 5$.

That is, a Diophantine quintuple is a set of $5$ positive integers such that the product of any $2$ of them plus $1$ is a square number.

Source of Name

This entry was named for Diophantus of Alexandria.

Sources

• 2015: Yifan Zhang and G. Grossman: On Diophantine triples and quadruples (Notes on Number Theory and Discrete Mathematics Vol. 21, no. 4: pp. 6 – 16)