العربية
中国
Français
Deutsch
Ελληνική
हिन्दी
Italiano
日本語
Português
Русский
EspañolFIBONACCI PRIME
A 'Fibonacci prime' is a Fibonacci number that is prime.
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 F''n'' 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 has been found probable primes for
:''n'' = 104911, 130021, 148091, 201107, 397379, 433781, 590041, 593689, 604711
Except for the case ''n'' = 4, if F''n'' is prime then ''n'' is prime. The converse is false, however.
F''p'' is prime for 8 out of the first 10 primes; the exceptions are F2 = 1 and F19 = 4181 = 37 × 113. However, Fibonacci primes become rarer as the index increases - F''p'' is prime for only 25 of the 1,229 primes ''p'' below 10,000.[1]
As of 2006, the largest known certain Fibonacci prime is F81839, with 17103 digits.[2] The largest known probable Fibonacci prime is F604711. It has 126377 digits and was found by Henri Lifchitz in 2005.[3]
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).[4]
For n≥3, Fn divides Fm iff n divides m.[5]
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.
1. Sloane's A005478, Sloane's A001605
2. Number Theory Archives announcement by David Broadhurst and Bouk de Water
3. PRP Records
4. Paulo Ribenboim, ''My Numbers, My Friends'', Springer-Verlag 2000
5. Wells 1986, p.65
★ Lucas number
★
★ R. Knott ''Fibonacci primes''
★ Caldwell, Chris. Fibonacci number, Fibonacci prime, and Record Fibonacci primes at the Prime Pages
The first Fibonacci primes are :
:2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, ...
| Contents |
| Known Fibonacci primes |
| Divisibility of Fibonacci numbers |
| References |
| See also |
| External links |
Known Fibonacci primes
It is not known if there are infinitely many Fibonacci primes. The first 33 are F''n'' 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 has been found probable primes for
:''n'' = 104911, 130021, 148091, 201107, 397379, 433781, 590041, 593689, 604711
Except for the case ''n'' = 4, if F''n'' is prime then ''n'' is prime. The converse is false, however.
F''p'' is prime for 8 out of the first 10 primes; the exceptions are F2 = 1 and F19 = 4181 = 37 × 113. However, Fibonacci primes become rarer as the index increases - F''p'' is prime for only 25 of the 1,229 primes ''p'' below 10,000.[1]
As of 2006, the largest known certain Fibonacci prime is F81839, with 17103 digits.[2] The largest known probable Fibonacci prime is F604711. It has 126377 digits and was found by Henri Lifchitz in 2005.[3]
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).[4]
For n≥3, Fn divides Fm iff n divides m.[5]
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.
References
1. Sloane's A005478, Sloane's A001605
2. Number Theory Archives announcement by David Broadhurst and Bouk de Water
3. PRP Records
4. Paulo Ribenboim, ''My Numbers, My Friends'', Springer-Verlag 2000
5. Wells 1986, p.65
See also
★ Lucas number
External links
★
★ R. Knott ''Fibonacci primes''
★ Caldwell, Chris. Fibonacci number, Fibonacci prime, and Record Fibonacci primes at the Prime Pages
This article provided by Wikipedia. To edit the contents of this article, click here for original source.
psst.. try this: add to faves
