The regularity of Diophantine quadruples

A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by the unity is a perfect square. A conjecture on the regularity of Diophantine quadruples asserts that any Diophantine triple can be uniquely extended to a Diophantine quadruple by joining an element exceeding the maximal element of the triple. The problem is reduced to studying an equation expressed as the coincidence of two linear recurrence sequences with initial terms composed of the fundamental solutions of some Pellian equations. In this paper, we determine the values of those initial terms completely and obtain finiteness results on the number of solutions of the equation. As one of the applications to the problem on the regularity of Diophantine quadruples, we show in general that the number of ways of extending any given Diophantine triple is at most $11$.