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

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*) ∈ **Z**^{2} and
let (*x*_{n})_{n ≥ 0}
be the sequence defined by some initial values
*x*_{0} and
*x*_{1} and the second-order linear recurrence
*x*_{n+1} =
*a**x*_{n} + *b**x*_{n−1}
for *n* ≥ 1. Suppose that *b* ≠ 0 and (*a*,*b*) ≠ (2,−1),(−2,−1).
Then there exist two relatively prime positive integers *x*_{0},
*x*_{1} such that |*x*_{n}| 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).

The above theorem extends a result of Graham, who solved the problem when (

(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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