Generalized Diophantine m-Tuples: A Survey

This survey article deals with the Diophantine m-tuples in positive integers as well as in some commutative rings with unity. In the first section, we give a brief history of the Diophantine m-tuples with the property D(n). We focus on these two problems: how many ways we can extend a Diophantine pair or triple to Diophantine quadruple; and how large a Diophantine m-tuple can be. In the last section, we generalize the concept of Diophantine m-tuples from positive integers to any commutative ring of unity and discuss some results on the line of the above problems.