On the number of extensions of a Diophantine triple

A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by unity is a perfect square. Any Diophantine triple is conjectured to be uniquely extended to a Diophantine quadruple by joining an element exceeding the maximal element in the triple. A previous work of the second and third authors revealed that the number of such extensions for a fixed Diophantine triple is at most 11. In this...