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On Pairwise Intersections of the Fibonacci, Sierpiński, and Riesel Sequences
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Dan Ismailescu

Department of Mathematics

Hofstra University

103 Hofstra University

Hempstead, NY 11549

USA

Peter Seho Park

Korea International School

373-6 Baekhyeon-dong, Budang-gu

Seongnam-si, Gyonggi-do

Korea

**Abstract:**

A *Sierpiński* number is an odd integer *k* with the property
that *k* · 2^{n} + 1 is
composite for all positive integer values of
*n*. A *Riesel* number is defined similarly; the only difference is
that *k* · 2^{n} - 1
is composite for all positive integer values of
*n*.
In this paper we find Sierpiński and Riesel numbers among the terms
of the well-known Fibonacci sequence. These numbers are smaller than
all previously constructed examples. We also find a 23-digit number
which is simultaneously a Sierpiński and a Riesel number. This
improves on the current record established by Filaseta, Finch and Kozek
in 2008. Finally, we prove that there are infinitely many values of *n*
such that the Fibonacci numbers *F*_{n}
and *F*_{n+1} are both
Sierpiński numbers.

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(Concerned with sequences
A000045
A076336
A076337.)

Received June 1 2013;
revised version received December 4 2013.
Published in *Journal of Integer Sequences*, December 5 2013.

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