Fibonacci prime
Encyclopedia
A Fibonacci prime is a Fibonacci number
that is prime
, a type of integer sequence prime
.
The first Fibonacci primes are :
In addition to these proven Fibonacci primes, there have been found probable prime
s for
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.
, 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.
(This implies the infinitude of primes
.)
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.
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.
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\; ....
that is 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...
, a type of integer sequence prime
Integer sequence prime
In mathematics, an integer sequence prime is a prime number found as a member of an integer sequence. For example, the 8th Delannoy number, 265729, is prime...
.
The first Fibonacci primes are :
- 2, 3, 5, 1313 (number)13 is the natural number after 12 and before 14. It is the smallest number with eight letters in its name spelled out in English. It is also the first of the teens – the numbers 13 through 19 – the ages of teenagers....
, 89, 233233 (number)233 is the natural number between 232 and 234. It is also a prime number.-In mathematics:233 is an irregular prime, a full reptend prime, a cousin prime, a Chen prime, a Fibonacci prime and a sexy prime. It is the 13th Fibonacci number. It is an Eisenstein prime of the form 3n - 1 with no...
, 1597, 28657, 514229, 433494437, 2971215073, ....
Known Fibonacci primes
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 prime
Probable prime
In number theory, a probable prime is an integer that satisfies a specific condition also satisfied by all prime numbers. Different types of probable primes have different specific conditions...
s 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 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.
, 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.
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).
(This implies the infinitude of primes
Euclid's theorem
Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. There are several well-known proofs of the theorem.-Euclid's proof:...
.)
For n ≥ 3, Fn divides Fm iff
IFF
IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...
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
Carmichael's theorem
Carmichael's theorem, named after the American mathematician R.D. Carmichael, states that for n greater than 12, the nth Fibonacci number F 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:Carmichael's theorem, named after...
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.
External links
- R. Knott Fibonacci primes
- Caldwell, Chris. Fibonacci number, Fibonacci prime, and Record Fibonacci primes at the Prime PagesPrime pagesThe Prime Pages is a website about prime numbers maintained by Chris Caldwell at the University of Tennessee at Martin.The site maintains the list of the "5,000 largest known primes", selected smaller primes of special forms, and many "top twenty" lists for primes of various forms...
- Small parallel Haskell program to find probable Fibonacci primes at haskell.org