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\vskip 1cm{\LARGE\bf Greatest Common Divisors in \\
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Shifted Fibonacci Sequences
}
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\large
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.\\
\href{mailto:kwchen@tmue.edu.tw}{\tt kwchen@tmue.edu.tw} \\
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\begin{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 $a\neq\pm 1$.
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