Carmichael's theorem

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 U_n(P,Q) with relatively prime parameters P, Q and positive discriminant, an element U_n with n ≠ 1, 2, 6 has at least one prime divisor that does not divide any earlier one except the 12nd Fibonacci number F(12)=U₁₂(1, -1)=144 and its equivalent U₁₂(-1, -1)=-144.

In particular, 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.

Carmichael (1913, Theorem 21) proved this theorem. Recently, Yabuta (2001) gave a simple proof.

Given two coprime integers P and Q, such that ${\displaystyle D=P^{2}-4Q>0}$ and PQ ≠ 0, let U_n(P,Q) be the Lucas sequence of the first kind defined by

Then, for n ≠ 1, 2, 6, U_n(P,Q) has at least one prime divisor that does not divide any U_m(P,Q) with m < n, except U₁₂(1, -1)=F(12)=144, U₁₂(-1, -1)=-F(12)=-144. Such a prime p is called a characteristic factor or a primitive prime divisor of U_n(P,Q). Indeed, Carmichael showed a slightly stronger theorem: For n ≠ 1, 2, 6, U_n(P,Q) has at least one primitive prime divisor not dividing D except U₃(1, -2)=U₃(-1, -2)=3, U₅(1, -1)=U₅(-1, -1)=F(5)=5, U₁₂(1, -1)=F(12)=144, U₁₂(-1, -1)=-F(12)=-144.

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