Only finitely many Tribonacci Diophantine triples exist

Diophantine Triples and k-Generalized Fibonacci Sequences

“…Our result generalizes the results obtained in [13], [11] and [8], where this problem was treated for the cases k = 2 and k = 3. In [13] it was shown that there does not exist a triple of positive integers a, b, c such that ab + 1, ac + 1, bc + 1 are Fibonacci numbers.…”

“…Finally, we prove the following result, which generalizes Proposition 1 in [8]. Observe that the upper bound depends now on k.…”

“…In order to simplify the notation we shall from now onwards write F n instead of F (k) n ; we still mean the nth k-generalized Fibonacci number. The arguments in this section follow the arguments from [8]. We will show that if there are infinitely many solutions to (1), then all of them can be parametrized by finitely many expressions as given in (18) for c below.…”

“…This now contradicts the following lemma, which turns out to be slightly more involved than in the special case on Tribonacci numbers (cf. [8]). Proof.…”

“…In [13] it was shown that there does not exist a triple of positive integers a, b, c such that ab + 1, ac + 1, bc + 1 are Fibonacci numbers. In [11] it was shown that there is no Tribonacci Diophantine quadruple, that is a set of four positive integers {a 1 , a 2 , a 3 , a 4 } such that a i a j + 1 is a member of the Tribonacci sequence (3-generalized Fibonacci sequence) for 1 ≤ i < j ≤ 4, and in [8] it was proved that there are only finitely many Tribonacci Diophantine triples. In the current paper we prove the same result for all such triples having values in the sequence of k-generalized Fibonacci numbers.…”