On members of Lucas sequences which are products of Catalan numbers

Authors

Shanta Laishram, Indian Statistical Institute (Delhi Centre)
Florian Luca, University of the Witwatersrand, Johannesburg
Mark Sias, University of Johannesburg

Article Type

Research Article

Publication Title

International Journal of Number Theory

Abstract

We show that if {Un}n≥0 is a Lucas sequence, then the largest n suc that |Un| = Cm1Cm2⋯Cmk with 1 ≤ m1 ≤ m2 ≤⋯ ≤ mk, where Cm is the mth Catalan number satisfies n < 6500. In case the roots of the Lucas sequence are real, we have n {1, 2, 3, 4, 6, 8, 12}. As a 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.

First Page

1487

Last Page

1515

DOI

10.1142/S1793042121500457

Publication Date

7-1-2021

Comments

Open Access, Green

Recommended Citation

Laishram, Shanta; Luca, Florian; and Sias, Mark, "On members of Lucas sequences which are products of Catalan numbers" (2021). Journal Articles. 1901.
https://digitalcommons.isical.ac.in/journal-articles/1901