On Diophantine $m$-tuples and $D(n)$-sets (Analytic Number Theory and Related Areas)

Abstract

For a nonzero integer n, a set of distinct nonzero integers {a_{1} : a_{2} : a_{m}} such that a_{i}a_{j}+n is a perfect square for all 1leq i<jleq m_{:} is called a Diophantine m-tuple with the property D(n) or simply D(n)-set. Such sets have been studied bince the ancient times. In this article, we give an overview of the results from the literature about D(n)-betb and summarize our recent findings about triples of integers which aleD(n)-sets for several n^{:}s. Furthermore, we include some new observations and remarks about the ways to construct such triples.