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 12th 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)^[1] gave a simple proof.

Given two relatively prime 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

${\displaystyle {\begin{aligned}U_{0}(P,Q)&=0,\\U_{1}(P,Q)&=1,\\U_{n}(P,Q)&=P\cdot U_{n-1}(P,Q)-Q\cdot U_{n-2}(P,Q)\qquad {\mbox{ for }}n>1.\end{aligned}}}$

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^[2] 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.

Note that D should be greater than 0; thus the cases U₁₃(1, 2), U₁₈(1, 2) and U₃₀(1, 2), etc. are not included, since in this case D = −7 < 0.

The only exceptions in Fibonacci case 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))

The smallest primitive prime divisor of F(n) are

1, 1, 2, 3, 5, 1, 13, 7, 17, 11, 89, 1, 233, 29, 61, 47, 1597, 19, 37, 41, 421, 199, 28657, 23, 3001, 521, 53, 281, 514229, 31, 557, 2207, 19801, 3571, 141961, 107, 73, 9349, 135721, 2161, 2789, 211, 433494437, 43, 109441, ... (sequence A001578 in the OEIS)

Carmichael's theorem says that every Fibonacci number, apart from the exceptions listed above, has at least one primitive prime divisor.

If n > 1, then the nth Pell number has at least one prime divisor that does not divide any earlier Pell number. The smallest primitive prime divisor of nth Pell number are

1, 2, 5, 3, 29, 7, 13, 17, 197, 41, 5741, 11, 33461, 239, 269, 577, 137, 199, 37, 19, 45697, 23, 229, 1153, 1549, 79, 53, 113, 44560482149, 31, 61, 665857, 52734529, 103, 1800193921, 73, 593, 9369319, 389, 241, ... (sequence A246556 in the OEIS)

• Zsigmondy's theorem

• Carmichael, R. D. (1913), "On the numerical factors of the arithmetic forms αⁿ±βⁿ", Annals of Mathematics, 15 (1/4): 30–70, doi:10.2307/1967797, JSTOR 1967797.