Diophantine Triples and k-Generalized Fibonacci Sequences
Diophantine Triples and k-Generalized Fibonacci Sequences
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 …