Periodica Mathematica Hungarica Vol. 47 (1–2), 2003, pp. 95–110 COMMON FACTORS OF SHIFTED FIBONACCI NUMBERS Santos Hern´ andez (Morelia) and Florian Luca (Morelia) [Communicated by: Attila Peth˝o] Abstract For any positive integer n let Fn be the nth Fibonacci number. Given positive integers a and b, we study the size of the greatest common divisor of Fn + a and Fm + b for varying positive integers m and n. Background and motivation Let P be a fixed finite set of prime numbers and let S be the set of all non- zero integers whose prime factors belong to P . Recently, the problem of finding non-trivial upper bounds for the greatest comon divisor of two expressions of the form x 1 + y 1 and x 2 + y 2 , where x 1 ,x 2 ,y 1 , and y 2 are numbers in S has received considerable interest. For example, the main result of [1] is that if a and b are two non-zero multiplicatively independent integers, then for every ε> 0 there exists a constant c ε so that whenever n>c ε is a positive integer then the inequality gcd(a n - 1,b n - 1) < exp(εn) holds. Of course, the condition that a and b are multiplicatively independent is necessary in order for the above conclusion to hold. The result from [1] was extended to the setting in which a n and b n are replaced by two multiplicatively independent elements x and y from S , and the above inequality was shown to hold in this setting as well when |x| and |y| are sufficiently large in [5] as well as in [7] (for the general case), and this result was enough to confirm a conjecture of Gy˝ ory, S´ ark˝ozy and Stewart from [6] concerning the largest prime factor of expressions of the form (ab + 1)(ac + 1)(bc + 1) in distinct positive integers a, b, and c. In a different direction, but of a somewhat similar flavour, a lot of work has been done concerning the problem of determining upper bounds for the greatest common divisor of two members of a non-degenerate binary recurrent sequence of integers (u n ) n≥0 . Recall that a non-degenerate binary recurrent sequence of integers (u n ) n≥0 is a sequence of integers such that there exist two non-zero integers r and s with r 2 +4s = 0 such that the recurrence formula u n+2 = ru n+1 + su n holds for Mathematics subject classification number: 11B39, 11J68. Key words and phrases: Fibonacci numbers, Euclidean algorithm, subspace theorem. 0031-5303/03/$20.00 Akad´ emiai Kiad´o, Budapest c  Akad´ emiai Kiad´o, Budapest Kluwer Academic Publishers, Dordrecht