Carmichael's Theorem

Carmichael's theorem, named after the American mathematician R.D. Carmichael, states that for n greater than 12, the nth Fibonacci number F(n) has at least one prime divisor that does not divide any earlier Fibonacci number.

The only exceptions for n up to 12 are:

F(1)=1 and F(2)=1, which have no prime divisors

F(6)=8 whose only prime divisor is 2 (which is F(3))

F(12)=144 whose only prime divisors are 2 (which is F(3)) and 3 (which is F(4))

If a prime p is a divisor of F(n) that does not divide any F(m) with m < n, then p is called a characteristic factor or a primitive prime divisor of F(n). Carmichael's theorem says that every Fibonacci number, apart from the exceptions listed above, has at least one primitive prime divisor.

The theorem can be generalized from Fibonacci numbers to other Lucas sequences.