Carmichael's theorem

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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 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 smaller n 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 of F(n). Carmichael's theorem says that every Fibonacci number, apart from the exceptions listed above, has at least one characteristic factor.

[edit] References

• R.D. Carmichael, On the numerical factors of the arithmetic forms αⁿ+βⁿ, Annals of Mathematics, 15, 1913, pp. 30–70.
• Ron Knott, The Mathematical Magic of the Fibonacci Numbers