Carmichael's theorem

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 "n"th 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.

References

*citation
last = Carmichael | first = R. D. | author-link = Robert Daniel Carmichael
doi = 10.2307/1967797
issue = 1/4
journal = Annals of Mathematics
pages = 30–70
title = On the numerical factors of the arithmetic forms α"n"+β"n"
volume = 15
year = 1913.

*citation
last = Knott | first = R.
publisher = [http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html Fibonacci Numbers and the Golden Section]
title = Fibonacci numbers and special prime factors
url = http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibmaths.html#primefactor.

*citation
last = Yabuta | first = M.
journal = Fibonacci Quarterly
pages = 493–443
title = A simple proof of Carmichael's theorem on primitive divisors
volume = 39
year = 2001.


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