# Diophantine Triples and k-Generalized Fibonacci Sequences

Bulletin of the Malaysian Mathematical Sciences Society, Aug 2016

We show that if $k\ge 2$ is an integer and $\big (F_n^{(k)}\big )_{n\ge 0}$ 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 $\big \{F_n^{(k)}: n\ge 1\big \}$. This generalizes a previous result where the statement for $k=3$ was proved. The result is ineffective since it is based on Schmidt’s subspace theorem.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs40840-016-0405-4.pdf

Clemens Fuchs, Christoph Hutle, Florian Luca, László Szalay. Diophantine Triples and k-Generalized Fibonacci Sequences, Bulletin of the Malaysian Mathematical Sciences Society, 2016, 1-17, DOI: 10.1007/s40840-016-0405-4