Fibonacci prime
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A Fibonacci prime is a Fibonacci number that is prime. The first few Fibonacci primes are
Except for the case n = 4, if Fn is prime then n is prime. The converse is false, however.
Fp is prime for 8 out of the first 10 primes; the exceptions are F2 = 1 and F19 = 4181 = 37 x 113. However, Fibonacci primes become rarer as the index increases - Fp is prime for only 25 of the 1,229 primes p below 10,000.[Sloane's A005478], [Sloane's A001605]
Currently, the largest known certain Fibonacci prime is F81839, with 17103 digits[Number Theory Archives announcement by David Broadhurst and Bouk de Water]; the largest known probable Fibonacci prime is F604711, with 126377 digits.[PRP Records]
It is not known if there are infinitely many Fibonacci primes.
Fibonacci numbers that have a prime index p do not share
any common divisors greater than 1 with the preceding
Fibonacci numbers, due to the identity
GCD(Fn, Fm) = FGCD(n,m).Paulo Ribenboim, My Numbers, My Friends, Springer-Verlag 2000
For n≥3, Fn divides Fm iff n divides m.
The greatest common divisor between any two Fibonacci
numbers, is equal to a Fibonacci number, with an index
that is the greatest common divisor, of the index values
held by the two Fibonacci numbers.
If we suppose that m, is a prime number p from the identity above,
and n is less than p, then it is clear that Fp, cannot
share any common divisors with the preceding Fibonacci
numbers.
GCD(Fp, Fn) = FGCD(p,n) = F1 = 1
Carmichael's theorem states that every Fibonacci
number (with a small set of exceptions) has at least one unique prime
factor that has not been a factor of the preceding
Fibonacci numbers
From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.Divisibility of Fibonacci numbers
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