# On members of Lucas sequences which are products of factorials

## TL;DR

This paper investigates Lucas sequences and proves that the largest index with a product of factorials as its term is less than 300,000, providing improved bounds for real-rooted sequences.

## Contribution

It establishes an upper bound on the index of Lucas sequence terms that are products of factorials, with tighter bounds for sequences with real roots.

## Key findings

- Largest such index n < 300,000 for general Lucas sequences
- Improved bounds for Lucas sequences with real roots
- Characterization of Lucas sequence terms as factorial products

## Abstract

Here, we show that if $\{U_n\}_{n\ge 0}$ is a Lucas sequence, then the largest $n$ such that $|U_n|=m_1!m_2!\cdots m_k!$ with $1<m_1\le m_2\le \cdots\le m_k$ satisfies $n<3\times 10^5$. We also give better bounds in case the roots of the Lucas sequence are real.