divides if and only if p is congruent to ±1 modulo 5, and p divides if and only if it is congruent to ±2 modulo 5. (For p = 5, F5 = 5 so 5 divides F5) Fibonacci numbers that have a prime index p do not share any common divisors greater than 1 with the preceding Fibonacci numbers, due to the identity: For , Fn divides Fm if and only if n divides m. If we suppose that m is a prime number p, and n is less than p, then it is clear that Fp cannot share any common divisors with the preceding Fibonacci numbers. This means that Fp will always have characteristic factors or be a prime characteristic factor itself. The number of distinct prime factors of each Fibonacci number can be put into simple terms. Fnk is a multiple of Fk for all values of n and k such that n ≥ 1 and k ≥ 1. It's safe to say that Fnk will have "at least" the same number of distinct prime factors as Fk. All Fp will have no factors of Fk, but "at least" one new characteristic prime from Carmichael's theorem. Carmichael's Theorem applies to all Fibonacci numbers except four special cases: and If we look at the prime factors of a Fibonacci number, there will be at least one of them that has never before appeared as a factor in any earlier Fibonacci number. Let πn be the number of distinct prime factors of Fn. If k | n then except for If k = 1, and n is an odd prime, then 1 | p and The first step in finding the characteristic quotient of any Fn is to divide out the prime factors of all earlier Fibonacci numbers Fk for which k | n. The exact quotients left over are prime factors that have not yet appeared. If p and q are both primes, then all factors of Fpq are characteristic, except for those of Fp and Fq. Therefore: The number of distinct prime factors of the Fibonacci numbers with a prime index is directly relevant to the counting function. Rank of apparition For a prime p, the smallest index u > 0 such that Fu is divisible by p is called the rank of apparition (sometimes called Fibonacci entry point) of p and denoted a(p). The rank of apparition