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The first Fibonacci primes are :
:2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, ....
It is not known if there are infinitely many Fibonacci primes. The first 33 are Fn for the n values : :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.
Except for the case n = 4, all Fibonacci primes have a prime index, but not all prime indexes are a Fibonacci prime.
Fp is prime for 8 out 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.
, the largest known certain Fibonacci prime is F81839, with 17103 digits. It was proved prime by David Broadhurst and Bouk de Water in 2001. The largest known probable Fibonacci prime is F1968721. It has 411439 digits and was found by Henri Lifchitz in 2009.
For n≥3, Fn divides Fm iff n divides m.
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 unique prime factor that has not been a factor of the preceding Fibonacci numbers.
Category:Classes of prime numbers Category:Fibonacci numbers Category:Unsolved problems in mathematics
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