Carmichael's theorem
- This article refers to Carmichael's theorem about Fibonacci numbers. Carmichael's theorem may also refer to the recursive definition of the Carmichael function.
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 factor that is not a factor of any earlier Fibonacci number. The only exceptions for n up to 12 are:
- F(1)=1 and F(2)=1, which have no prime factors
- F(6)=8 whose only prime factor is 2 (which is F(3))
- F(12)=144 whose only prime factors are 2 (which is F(3)) and 3 (which is F(4))
If a prime p is a factor of F(n) and not a factor of any F(m) with m < n then p is called a characteristic factor or a primitive divisor of F(n). Carmichael's theorem says that every Fibonacci number, apart from the exceptions listed above, has at least one characteristic factor.
The theorem can be generalized from Fibonacci numbers to other Lucas sequences.
References
- Carmichael, R. D. (1913), "On the numerical factors of the arithmetic forms αn+βn", Annals of Mathematics, 15 (1/4), Annals of Mathematics: 30–70, doi:10.2307/1967797, JSTOR 1967797.
- Knott, R., Fibonacci numbers and special prime factors, Fibonacci Numbers and the Golden Section
{{citation}}: External link in|publisher= - Yabuta, M. (2001), "A simple proof of Carmichael's theorem on primitive divisors" (PDF), Fibonacci Quarterly, 39: 439–443.