Fibonacci prime
A Fibonacci prime is a Fibonacci number that is prime, a type of integer sequence prime.
The first Fibonacci primes are (sequence A005478 in OEIS):
Contents |
[edit] Known Fibonacci primes
| Are there an infinite number of Fibonacci primes? |  |
It is not known if there are infinitely many Fibonacci primes. The first 33 are Fn for the n values (sequence A001605 in OEIS):
- 3, 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, 9311, 9677, 14431, 25561, 30757, 35999, 37511, 50833, 81839.
In addition to these proven Fibonacci primes, there have been found probable primes for
- n = 104911, 130021, 148091, 201107, 397379, 433781, 590041, 593689, 604711, 931517, 1049897, 1285607, 1636007, 1803059, 1968721.[1]
Except for the case n = 4, all Fibonacci primes have a prime index, but not every prime is the index of a Fibonacci prime.
Fp is prime for 8 of the first 10 primes p; the exceptions are F2 = 1 and F19 = 4181 = 37 × 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.[2]
As of November 2009[update], the largest known certain Fibonacci prime is F81839, with 17103 digits. It was proved prime by David Broadhurst and Bouk de Water in 2001.[3][4] The largest known probable Fibonacci prime is F1968721. It has 411439 digits and was found by Henri Lifchitz in 2009.[1]
[edit] Divisibility of Fibonacci numbers
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).[5]
(This implies the infinitude of primes.)
For n ≥ 3, Fn divides Fm iff n divides m.[6]
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 (except for 1, 8 and 144) has at least one prime factor that has not been a factor of the preceding Fibonacci numbers.
[edit] See also
[edit] References
- ^ a b PRP Top Records, Search for : F(n). Retrieved 2009-11-21.
- ^ Sloane's
A005478, A001605 - ^ Number Theory Archives announcement by David Broadhurst and Bouk de Water
- ^ Chris Caldwell, The Top Twenty: Fibonacci Number from the Prime Pages. Retrieved 2009-11-21.
- ^ Paulo Ribenboim, My Numbers, My Friends, Springer-Verlag 2000
- ^ Wells 1986, p.65
[edit] External links
- Weisstein, Eric W., "Fibonacci Prime" from MathWorld.
- R. Knott Fibonacci primes
- Caldwell, Chris. Fibonacci number, Fibonacci prime, and Record Fibonacci primes at the Prime Pages
- Small parallel Haskell program to find probable Fibonacci primes at haskell.org
