Tribonacci Numbers That Are Products of Two Lucas Numbers

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.

Theorems & Definitions (12)

• Theorem 2.1
• Definition 3.1
• Lemma 3.2
• proof
• Definition 3.3
• Theorem 3.4: Matveev
• Lemma 3.5: Dujella--Pethö
• Lemma 3.6: sanchez2014linear
• Lemma 3.7: Ddamulira--Luca--Rakotomalala
• proof