What is a Diophantine $m$-tuple?

Abstract

The problem of the construction of Diophantine m-tuples, i.e. sets with the property that product of any two of its distinct elements is one less then a square, has a very long history. There are some new results in this area, but many open problems and unproved conjectures still remains. In this survey we explain the main problems and results concerning Diophantine m-tuples.