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: Leonhard Euler is considered by many people to be one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is mathematics. ... Robert Daniel Carmichael (1879-1967) was a leading American mathematician. ... In mathematics, the Fibonacci numbers, named after Leonardo of Pisa, known as Fibonacci, form a sequence defined recursively by: In other words,each number is the sum of the two numbers before it. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ...

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. A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ...

References

R.D. Carmichael, On the numerical factors of the arithmetic forms αⁿ+βⁿ, Annals of Mathematics, 15, 1913, pp. 30–70.