Only finitely many Tribonacci Diophantine triples exist

TL;DR

This paper proves that only finitely many integer triples exist where the pairwise products plus one are Tribonacci numbers, using the Subspace theorem, thus answering a recent open question in recurrence sequence Diophantine problems.

Contribution

It establishes the finiteness of such Tribonacci Diophantine triples, a problem previously unresolved, employing the Subspace theorem for the proof.

Findings

Finitely many Tribonacci Diophantine triples exist

The proof relies on the Subspace theorem

Addresses an open question in recurrence sequence Diophantine analysis

Abstract

Diophantine triples taking values in recurrence sequences have recently been studied quite a lot. In particular the question was raised whether or not there are finitely many Diophantine triples in the Tribonacci sequence. We answer this question here in the affirmative. We prove that there are only finitely many triples of integers $1\le u<v<w$ such that $uv+1,uw+1,vw+1$ are Tribonacci numbers. The proof depends on the Subspace theorem.