Fibonacci sequence is defined by f(1)=1,f(2)=1 and f(m) =f(m-1)+f(m-2)
i.e. 1, 1, 2, 3, 5, 8, 13, ...

Prove that for all composite values of n>4, f(n) is composite.

Ady's new problem seems to ask for proof that IF index n is composite (i.e. not prime), THEN F(n) is also composite (not prime). This seems the contrapositive formulatiion of Carmichael's theorem that IF F(n) is prime (not composite) THEN index n must be prime (not composite) -- with the same restriction to exclude F(4)=3, in all cases. Neither is asserting that if n is prime, then F(n) must be prime -- which is in any case false (e.g. F(19)=4181=37*113.) Is there another twist?