# Definition:Diophantine m-Tuple

## Contents

## 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*: 6 – 16)