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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 where the statement for \(k=3\) was proved. The result ...