Fibonacci prime

Fibonacci prime

A Fibonacci prime is a Fibonacci number that is prime.

The first Fibonacci primes are OEIS|id=A005478:

:2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, ....

Known Fibonacci primes

It is not known if there are infinitely many Fibonacci primes. The first 33 are F"n" for the "n" values OEIS2C|id=A001605::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.

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 "p"; 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. [ [http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A005478 Sloane's A005478] , [http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001605 Sloane's A001605] ]

As of 2006, the largest known certain Fibonacci prime is F81839, with 17103 digits. [ [http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0104&L=nmbrthry&P=R1807&D=0 Number Theory Archives announcement by David Broadhurst and Bouk de Water] ] The largest known probable Fibonacci prime is F604711. It has 126377 digits and was found by Henri Lifchitz in 2005. [ [http://www.primenumbers.net/prptop/prptop.php PRP Records] ]

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). [Paulo Ribenboim, "My Numbers, My Friends", Springer-Verlag 2000]

For n≥3, Fn divides Fm iff n divides m. [Wells 1986, p.65]

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.

ee also

* Lucas number

References

External links

*
* [http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibmaths.html#fibprimes R. Knott "Fibonacci primes"]
* Caldwell, Chris. [http://primes.utm.edu/glossary/page.php/FibonacciNumber.html Fibonacci number] , [http://primes.utm.edu/glossary/page.php?sort=FibonacciPrime Fibonacci prime] , and [http://primes.utm.edu/top20/page.php?id=39 Record Fibonacci primes] at the Prime Pages
* Small parallel Haskell program to find probable Fibonacci primes at [http://www.haskell.org/haskellwiki/Fibonacci_primes_in_parallel haskell.org]


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Fibonacci number — A tiling with squares whose sides are successive Fibonacci numbers in length …   Wikipedia

  • Fibonacci — Infobox Scientist box width = 300px name = Leonardo of Pisa (Fibonacci) image width = 150px caption = Leonardo of Pisa, Fibonacci birth date = c. 1170 birth place = Pisa, Italy death date = c. 1250 death place = Pisa, Italy residence = Italy… …   Wikipedia

  • Prime number — Prime redirects here. For other uses, see Prime (disambiguation). A prime number (or a prime) 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… …   Wikipedia

  • Fibonacci-Folge — Ein Kachelmuster aus Quadraten, deren Kantenlänge der Fibonacci Folge entspricht Die Fibonacci Folge ist eine unendliche Folge von Zahlen (den Fibonacci Zahlen), bei der sich die jeweils folgende Zahl durch Addition ihrer beiden vorherigen Zahlen …   Deutsch Wikipedia

  • Fibonacci-Reihe — Ein Kachelmuster aus Quadraten, deren Kantenlänge der Fiboncci Folge entspricht Die Fibonacci Folge ist eine unendliche Folge von Zahlen (den Fibonacci Zahlen), bei der sich die jeweils folgende Zahl durch Addition der beiden vorherigen Zahlen… …   Deutsch Wikipedia

  • Fibonacci-Zahl — Ein Kachelmuster aus Quadraten, deren Kantenlänge der Fiboncci Folge entspricht Die Fibonacci Folge ist eine unendliche Folge von Zahlen (den Fibonacci Zahlen), bei der sich die jeweils folgende Zahl durch Addition der beiden vorherigen Zahlen… …   Deutsch Wikipedia

  • Fibonacci-Zahlen — Ein Kachelmuster aus Quadraten, deren Kantenlänge der Fiboncci Folge entspricht Die Fibonacci Folge ist eine unendliche Folge von Zahlen (den Fibonacci Zahlen), bei der sich die jeweils folgende Zahl durch Addition der beiden vorherigen Zahlen… …   Deutsch Wikipedia

  • 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. A challenge in empirical mathematics is to identify large prime values in rapidly… …   Wikipedia

  • List of prime numbers — This is an incomplete list, which may never be able to satisfy particular standards for completeness. You can help by expanding it with reliably sourced entries. By Euclid s theorem, there are an infinite number of prime numbers. Subsets of the… …   Wikipedia

  • Suite de Fibonacci — La suite de Fibonacci est une suite d entiers très connue. Elle doit son nom à Leonardo Fibonacci, dit Leonardo Pisano, un mathématicien italien du XIIIe siècle qui, dans un problème récréatif posé dans un de ses ouvrages, le Liber Abaci,… …   Wikipédia en Français

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”