Calculate GCD(Fib(531185354674),Fib(613570636967))

Question

I picked this problem from here http://mathalon.in/?page=show_problem.php&pid=106 . I don't know how to solve it accurately. The number seems too to be done by any computation method.

Answer 1

Try and compute $\gcd(F_i,F_j)$ for some small $i$ and $j$ and see if you can spot a pattern.

Using the following three facts you can prove by induction that the pattern holds in general:

• $\gcd(F_n,F_{n-1}) = 1$.
• $F_{m+n} = F_{m+1}F_n + F_mF_{n-1}$.
• $m$ divides $n \implies F_m$ divides $F_n$.

Answer 2

GCD(Fib(531185354674),Fib(613570636967)) = Fib(GCD(531185354674,613570636967))