Sets with the Property \(D(n)\)

This chapter is devoted to the generalization of the notion of Diophantine m-tuples in which the condition that \(ab+1\) is a square is replaced by \(ab+n\) is a square for a fixed integer or rational number n. We discuss the problem of the existence of integer \(D(n)\) -quadruples and rational \(D(n)\) -quintuples. The last problem is related to the distribution of ranks in families of twists of certain elliptic curves. Finally, we consider sets which are \(D(n)\) -triples, quadruples and quintuples for several distinct values of n. Elliptic curves also play an important role here, but their appearance differs in those three problems.