On members of Lucas sequences which are products of factorials

Article Type

Research Article

Publication Title

Monatshefte fur Mathematik

Abstract

We show that if {Un}n≥0 is a Lucas sequence, then the largest n such that | Un| = m1! m2! ⋯ mk! with 1 ≤ m1≤ m2≤ ⋯ ≤ mk 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 {Xn}n≥1 is the sequence of X-coordinates of a Pell equation X2- dY2= ± 1 with a non-zero integer d> 1 , then Xn= m! implies n= 1.

First Page

329

Last Page

359

DOI

10.1007/s00605-020-01455-y

Publication Date

10-1-2020

Comments

Open Access, Green

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