An Absolute Bound for the Size of Diophantine m-Tuples Original Research Article

A set of m positive integers is called a Diophantine m-tuple if the product of its any two distinct elements increased by 1 is a perfect square. We prove that if {a, b, c} is a Diophantine triple such that b>4a and c>max{b13, 1020} or c>max{b5, 101029}, then there is unique positive integer d such that d>c and {a, b, c, d} is a Diophantine quadruple. Furthermore, we prove that there does not exist a Diophantine 9-tuple and that there are only finitely many Diophantine 8-tuples.