In number theory, Carmichael's theorem, named after the American mathematician R. D. Carmichael, states that, for any nondegenerate Lucas sequence of the first kind Un(P, Q) with relatively prime parameters P, Q and positive discriminant, an element Un with n ≠ 1, 2, 6 has at least one prime divisor that does not divide any earlier one except the 12th Fibonacci number F(12) = U12(1, −1) = 144 and its equivalent U12(−1, −1) = −144.
Bilu, Hanrot, Voutier and Mignotte (2001)[2] extended it to the case of negative discriminants (where it is true for all n > 30).
and PQ ≠ 0, let Un(P, Q) be the Lucas sequence of the first kind defined by Then, for n ≠ 1, 2, 6, Un(P, Q) has at least one prime divisor that does not divide any Um(P, Q) with m < n, except U12(±1, −1) = ±F(12) = ±144.
Indeed, Carmichael showed a slightly stronger theorem: For n ≠ 1, 2, 6, Un(P, Q) has at least one primitive prime divisor not dividing D[3] except U3(±1, −2) = 3, U5(±1, −1) = F(5) = 5, or U12(1, −1) = −U12(−1, −1) = F(12) = 144.
The only exceptions in Fibonacci case for n up to 12 are: The smallest primitive prime divisor of F(n) are Carmichael's theorem says that every Fibonacci number, apart from the exceptions listed above, has at least one primitive prime divisor.