SAMS Subject Classification Number: 22

We determine upper bounds on \(n\) when the \(n\)th term of a Lucas sequence is expressible as a product of factorials. As our starting point we use an adaptation of the proof of the Primitive Divisor Theorem. Firstly, a series of lemmas are proven as a means to establishing an upper bound on \(n.\) Some of these proofs exploit sieve methods, the properties of cyclotomic polynomials, and results related to linear forms in logarithms, among others. In fact, 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 \(n<62000\). When the roots of the Lucas sequence are real, we have \(n\in \{1,2, 3, 4, 6,12\}\). As a consequence, we formulate and prove a corollary regarding the \(X\)- coordinates of Pell equations which are products of factorials. We show that if \(\{X_n\}_{n\geq 1}\) is the sequence of \(X\)- coordinates of a Pell equation \(X^2-dY^2=\pm 1\) with a nonsquare integer \(d>1\), then \(X_n=m!\) implies \(n=1\).