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 Un 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=U12=144 and its equivalent U12=-144.
In particular, for n greater than 12, the nth Fibonacci number F has at least one prime divisor that does not divide any earlier Fibonacci number.
Carmichael proved this theorem. Recently, Yabuta gave a simple proof.

Statement

Given two coprime integers P and Q, such that and, let be the Lucas sequence of the first kind defined by
Then, for n ≠ 1, 2, 6, Un has at least one prime divisor that does not divide any Um with m < n, except U12=F=144, U12=-F=-144.
Such a prime p is called a characteristic factor or a primitive prime divisor of Un.
Indeed, Carmichael showed a slightly stronger theorem: For n ≠ 1, 2, 6, Un has at least one primitive prime divisor not dividing D except U3=U3=3, U5=U5=F=5, U12=F=144, U12=-F=-144.
Note that D should be > 0, thus the cases U13, U18 and U30, etc. are not included, since in this case D = −7 < 0.

Fibonacci and Pell cases

The only exceptions in Fibonacci case for n up to 12 are:
The smallest primitive prime divisor of F are
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