On members of Lucas sequences which are products of factorials

We show that if { U n } n ≥ 0 is a Lucas sequence, then the largest n such that | U n | = m 1 ! m 2 ! ⋯ m k ! with 1 ≤ m 1 ≤ m 2 ≤ ⋯ ≤ m k satisfies n < 62,000. When the roots of the Lucas sequence are real, we have n ∈ { 1 , 2 , 3 , 4 , 6 , 12 } . As a consequence, we show that if { X n } n ≥ 1 is the sequence of X -coordinates of a Pell equation X 2 - d Y 2 = ± 1 with a non-zero integer d > 1 , then X n = m ! implies n = 1 .