On a conjecture concerning the multiplicity of the Tribonacci sequence

On $k$-generalized Fibonacci numbers with negative indices

“…Much less is known if n or m is negative. In this direction the first step was done by Bravo, Gómez and Luca [4], see also Bravo et al [5], who solved completely the equation F Our Theorem 3.4 is much weaker than Marques's above cited result. The reason is that for negative indices the dominating root is not large enough, and all but one roots are lying outside the unit circle.…”

“…Bravo, Gómez and Luca [4] established all solutions of T (3)m = 0, m ∈ Z, i.e., all zero terms in the Tribonacci sequence. Bravo et al[5] proved that there are only eight integers, which appear at least twice in the Tribonacci sequence, and computed all solutions of T(3) m = c in the remaining eight cases. Here we prove Theorem 4.3.…”

Common values of a class of linear recurrence

“…The next argumentation was used for the tribonacci numbers by Bravo et al [4,3] (especially the proof of Lemma 1 in [3]). The zeroes of the quadratic polynomial…”

“…Recently E.F. Bravo et al [4], [3] established all solutions n, m ∈ Z of the diophantine equation T n = T m , where (T n ) denotes the tribonacci sequence, which is defined by the initial terms T −1 = T 0 = 0, T 1 = 1 and by the recursion…”

“…As I mentioned Bravo et al were interested to find all common terms for a given recurrence, but after a closer look became clear that the method is capable to prove effective finiteness result for a wide class of recurrences. My original goal was to generalize the result of E.F. Bravo et al [4], [3] to common values of kgeneralized Fibonacci numbers, see e.g. [10].…”