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Greatest Common Divisors of Shifted Fibonacci Sequences Revisited
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Annalena Rahn and Martin Kreh

Institute of Mathematics and Applied Computer Science

University of Hildesheim

Samelsonplatz 1

31141 Hildesheim

Germany

**Abstract:**

In 2011, Chen computed the greatest common divisors of consecutive
shifted Fibonacci numbers *F*_{n} + *a* and
*F*_{n+1} + *a* for *a* ∈ {±1,
±2}. He also showed that gcd(*F*_{n} + *a*,
*F*_{n+1} + *a*) is bounded if *a* ≠ ±1.
This was later generalized by Spilker, who also showed that
gcd(*F*_{n} + *a*,
*F*_{n+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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