On members of Lucas sequences which are products of factorials

Abstract

We show that if{Un}n≥0is a Lucas sequence, then the largest n such that |Un|=m1!m2!···mk! with 1≤m1≤m2≤···≤mk satisfies n < 62000. 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 {Xn}n≥1is the sequence of X coordinates of a Pell equationX2−dY2=±1 with a nonsquare integer d >1, then Xn=m! implies n= 1. Moreover, we show that the largest n such that |Un|=Cm1Cm2···Cmkwith1≤m1≤m2≤···≤mk, where Cm is the mth Catalan number satisfies n < 6500. In case the roots of the Lucas sequence are real, we haven∈{1,2,3,4,6,8,12}. Asa consequence, we show that if {Xn}n≥1 is the sequence of the X coordinates of a Pell equation X2−dY2=±1 with a nonsquare integer d >1, then Xn=Cm implies n= 1