Carmichael's theorem
Encyclopedia
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:
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 sequence
s.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
R.D. Carmichael
Robert Daniel Carmichael
Robert Daniel Carmichael was a leading American mathematician. Carmichael was born in Goodwater, Alabama. He attended Lineville College, briefly, and he earned his bachelor's degree in 1898, while he was studying towards his Ph.D. degree at Princeton University. Carmichael completed the...
, states that for n greater than 12, the nth Fibonacci number
Fibonacci number
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....
F(n) has at least one prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
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
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
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 sequence
Lucas sequence
In mathematics, the Lucas sequences Un and Vn are certain integer sequences that satisfy the recurrence relationwhere P and Q are fixed integers...
s.