Journal of Integer Sequences, Vol. 22 (2019), Article 19.7.2

On Second-Order Linear Sequences of Composite Numbers


Dan Ismailescu
Department of Mathematics
Hofstra University
Hempstead, NY 11549
USA

Adrienne Ko
Ethical Culture Fieldston School
Bronx, NY 10471
USA

Celine Lee
Chinese International School
Hong Kong SAR
China

Jae Yong Park
The Lawrenceville School
Lawrenceville, NJ 08648
USA

Abstract:

We present a new proof of the following result of Somer:
Let (a, b) ∈ Z2 and let (xn)n ≥ 0 be the sequence defined by some initial values x0 and x1 and the second-order linear recurrence xn+1 = axn + bxn−1 for n ≥ 1. Suppose that b ≠ 0 and (a,b) ≠ (2,−1),(−2,−1). Then there exist two relatively prime positive integers x0, x1 such that |xn| is a composite integer for all n ∈ N.
The above theorem extends a result of Graham, who solved the problem when (a, b) = (1, 1).

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(Concerned with sequences A001109 A001353 A001906 A004254.)

Received January 7 2019; revised versions received February 4 2019; September 8 2019; September 24 2019. Published in Journal of Integer Sequences, September 24 2019.

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