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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.
mailto:kwchen@tmue.edu.twkwchen@tmue.edu.tw

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Abstract:

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

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

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 $a\neq\pm 1$.