On the 𝑝-adic zeros of the Tribonacci sequence

Bilu, Yuri; Luca, Florian; Nieuwveld, Joris; Ouaknine, Joël; Worrell, James

doi:10.1090/mcom/3893

On the $p$-adic zeros of the Tribonacci sequence
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by Yuri Bilu, Florian Luca, Joris Nieuwveld, Joël Ouaknine and James Worrell

Math. Comp. 93 (2024), 1333-1353

DOI: https://doi.org/10.1090/mcom/3893

Published electronically: August 31, 2023

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Abstract:

Let $(T_n)_{n\in {\mathbb Z}}$ be the Tribonacci sequence and for a prime $p$ and an integer $m$ let $\nu _p(m)$ be the exponent of $p$ in the factorization of $m$. For $p=2$ Marques and Lengyel found some formulas relating $\nu _p(T_n)$ with $\nu _p(f(n))$ where $f(n)$ is some linear function of $n$ (which might be constant) according to the residue class of $n$ modulo $32$ and asked if similar formulas exist for other primes $p$. In this paper, we give an algorithm which tests whether for a given prime $p$ such formulas exist or not. When they exist, our algorithm computes these formulas. Some numerical results are presented.

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