Yet another $S$-unit variant of Diophantine tuples

Abstract

We show that there are only finitely many triples of integers $ 0 < a < b < c $ such that the product of any two of them is the value of a given polynomial with integer coefficients evaluated at an $ S $-unit that is also a positive integer. The proof is based on a result of Corvaja and Zannier and thus is ultimately a consequence of the Schmidt subspace theorem.