Diophantine triples with values in $k$-generalized Fibonacci sequences

TL;DR

This paper proves that for any fixed integer k≥2, there are only finitely many Diophantine triples with elements in the k-generalized Fibonacci sequence, extending previous results for the case k=3.

Contribution

The paper generalizes the finiteness result of Diophantine triples with Fibonacci-like sequence values to all k-generalized Fibonacci sequences for k≥2.

Findings

Finitely many such triples exist for each fixed k≥2

Extension of previous results from k=3 to general k

Uses Schmidt's subspace theorem to establish finiteness

Abstract

We show that if $k\ge 2$ is an integer and $(F_n^{(k)})_{n\ge 0}$ is the sequence of $k$-generalized Fibonacci numbers, then there are only finitely many triples of positive integers $1<a<b<c$ such that $ab+1,~ac+1,~bc+1$ are all members of $\{F_n^{(k)}: n\ge 1\}$. This generalizes a previous result (cf. arXiv:1508.07760) where the statement for $k=3$ was proved. The result is ineffective since it is based on Schmidt's subspace theorem.