Journal of Integer Sequences, Vol. 21 (2018), Article 18.6.7

Greatest Common Divisors of Shifted Fibonacci Sequences Revisited

Annalena Rahn and Martin Kreh
Institute of Mathematics and Applied Computer Science
University of Hildesheim
Samelsonplatz 1
31141 Hildesheim


In 2011, Chen computed the greatest common divisors of consecutive shifted Fibonacci numbers Fn + a and Fn+1 + a for a ∈ {±1, ±2}. He also showed that gcd(Fn + a, Fn+1 + a) is bounded if a ≠ ±1. This was later generalized by Spilker, who also showed that gcd(Fn + a, Fn+1 + a) is periodic if a ≠ ±1. In this article, we compute the greatest common divisor for a = ±3 and we show how the results given in this article compare to bounds derived by Chen and periods derived by Spilker. We further give a necessary criterion for an integer d to occur as such a greatest common divisor.

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(Concerned with sequence A000045.)

Received January 31 2018; revised version received June 20 2018. Published in Journal of Integer Sequences, August 22 2018.

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