Products of Tribonacci Numbers that are the Products of Factorials

TL;DR

This work extends the study of intersections between Tribonacci numbers and factorial products by solving two new classes of equations: products of consecutive Tribonacci terms equalling products of factorials, and the corresponding absolute-value products for negative indices. It employs a combination of Binet-type representations, $2$-adic valuations, and explicit computational checks to establish finiteness and identify all solutions in key cases. The main results are threefold: (i) a unique positive-index solution for consecutive products, (ii) a finite explicit list of negative-index solutions, and (iii) a general finiteness result with concrete bounds for the multi-step case with gap $d\ge2$, namely $rd\le1{,}439{,}862$ and $n<719{,}933$. These findings deepen the understanding of how higher-order recurrence sequences interact with products of factorials and set the stage for potential explicit enumerations in broader contexts.

Abstract

In 2014 Marques and Lengyel gave all of the solutions to the equation $T_n=m!$, where $T_n$ is the $n$th term of the Tribonacci sequence $0,1,1,2,4,7,13,24,\ldots$. In 2023 Alahmadi and Luca generalized their result to the equation $T_n=m_1!m_2!\cdots m_k!$ for every $k\in\mathbb{N}$, where $m_1\leq m_2\leq\ldots\leq m_k$ listing all the solutions to this equation. Here we generalize these results further and give all the solutions to $T_nT_{n+1}T_{n+2}\cdots T_{n+r}=m_1!m_2!\cdots m_k!$ and $ |T_{-n}T_{-n-1}T_{-n-2}\cdots T_{-n-r}|=m_1!m_2!\cdots m_k!$ for every $n,r\in\mathbb{N}$, where $m_1\leq m_2\leq\ldots\leq m_k$.

Theorems & Definitions (11)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Definition 2.4
• Definition 2.5
• Lemma 3.1
• proof : Proof of Lemma \ref{['first-bound']}
• Lemma 3.2
• proof
• proof