Tribonacci Numbers That Are Products of Two Lucas Numbers

TL;DR

This paper investigates the solutions to the Diophantine equation where Tribonacci numbers are expressed as products of two Lucas numbers, providing a complete solution and proof for all positive integers involved.

Contribution

It offers a complete characterization and proof of all solutions to the equation T_k = L_m * L_n involving Tribonacci and Lucas numbers.

Findings

Identifies all solutions to T_k = L_m * L_n for positive integers.

Proves the non-existence of solutions beyond certain bounds.

Provides a complete classification of solutions.

Abstract

Let $T_{k}$ be the $k^{\textrm{th}}$ Tribonacci number and $L_{n}$ be the $n^{\textrm{th}}$ Lucas number defined by their respective recurrence relation $T_{k}=T_{k-1}+T_{k-2}+T_{k-3}$ and $L_{n}=L_{n-1}+L_{n-2}$. In this study, we solve the Diophantine equation $T_{k} = L_{m}L_{n}$ for positive integer unknowns $m$, $n$, and $k$ and prove our results.