Type: Article

Diophantine triples and k-generalized Fibonacci sequences

Fuchs, Clemens; Hutle, Christoph; Luca, Florian; Szalay, László;

Abstract:

We show that if k ? 2 is an integer and F(k) n n?0 is the sequence of kgeneralized Fibonacci numbers, then there are only ?nitely many triples of positive integers 1 < a < b < c such that ab+ 1, ac+ 1, bc + 1 are all members of F(k) n :n ?1. 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.