Journal of Integer Sequences, Vol. 14 (2011), Article 11.4.7

Greatest Common Divisors in Shifted Fibonacci Sequences


Kwang-Wu Chen
Department of Mathematics and Computer Science Education
Taipei Municipal University of Education
No. 1, Ai-Kuo West Road
Taipei, Taiwan 100, R.O.C.

Abstract:

It is well known that successive members of the Fibonacci sequence are relatively prime. Let

$\displaystyle f_n(a)=\gcd(F_n+a,F_{n+1}+a). $

Therefore $ (f_n(0))$ is the constant sequence $ 1,1,1,\ldots$, but Hoggatt in 1971 noted that $ (f_n(\pm1))$ is unbounded. In this note we prove that $ (f_n(a))$ is bounded if $ a\neq\pm 1$.

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(Concerned with sequences A000032 A000045 A000071 A001611 A157725.)

Received January 31 2011; revised version received March 26 2011. Published in Journal of Integer Sequences, March 26 2011.

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