Fib 性质 Gcd(f[n],f[m]) = f(gcd(n,m))

a) gcd(fn, fn-1) = 1, for all n 
b) fm+n = fm+1 fn + fm fn-1 

c) if m divides n, then fm divides fn 

and the ever important Euclidean Algorithm which states: 
if n=qm+r, then gcd(n,m)=gcd(m,r). For such n,m we have
gcd(fm,fn) = gcd(fm,fqm+r) = gcd(fm,fqm+1fr+fqmfr-1) = gcd(fm,fqm+1fr) = gcd(fm,fr)
gcd(fn,fm)=gcd(fm,fr)

性质a,更相减损法即可证。
性质b,展开即显然。
由性质b和性质a，即可证得gcd(fm,fn) = fgcd(m,n).