Journal of Integer Sequences, Vol. 16 (2013), Article 13.9.8

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

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

Peter Seho Park
Korea International School
373-6 Baekhyeon-dong, Budang-gu
Seongnam-si, Gyonggi-do


A Sierpiński number is an odd integer k with the property that k · 2n + 1 is composite for all positive integer values of n. A Riesel number is defined similarly; the only difference is that k · 2n - 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 Fn and Fn+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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