Yet another $S$-unit variant of Diophantine tuples

TL;DR

The paper proves finiteness results for certain Diophantine triples involving products of integers and polynomial values at S-units, extending the understanding of such tuples using advanced number theory techniques.

Contribution

It introduces a new finiteness result for triples of integers related to polynomial values at S-units, utilizing the Schmidt subspace theorem.

Findings

Finiteness of triples with specified properties

Application of Corvaja and Zannier's result

Use of Schmidt subspace theorem

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.