On members of Lucas sequences which are products of factorials

Abstract

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 \(1\le m_1\le m_2\le \cdots \le m_k\) satisfies \(n<\) 62,000. When the roots of the Lucas sequence are real, we have \(n\in \{1, 2, 3, 4, 6, 12\}\). As a consequence, we show that if \(\{X_n\}_{n\ge 1}\) is the sequence of X-coordinates of a Pell equation \(X^2-dY^2=\pm 1\) with a non-zero integer \(d>1\), then \(X_n=m!\) implies \(n=1\).