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Fibonacci Prime

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A Fibonacci prime is a Fibonacci number F_n that is also a prime number. Every F_n that is prime must have a prime index n, with the exception of F_4=3. However, the converse is not true (i.e., not every prime index p gives a prime F_p).

The first few (possibly probable) prime Fibonacci numbers F_n are 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ... (OEIS A005478), corresponding to indices n=3, 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, ... (OEIS A001605). (Note that Gardner's statement that F_(531) is prime (Gardner 1979, p. 161) is incorrect, especially since 531 is not even prime, which it must be for F_(531) to be prime.) The following table summarizes Fibonacci (possibly probable) primes with index n>5000.

termindexdigitsdiscovererstatus
2453871126proven prime; http://primes.utm.edu/primes/page.php?id=51129
2593111946proven prime; http://primes.utm.edu/primes/page.php?id=37470
2696772023proven prime; http://primes.utm.edu/primes/page.php?id=35537
27144313016proven prime; http://primes.utm.edu/primes/page.php?id=29537
28255615342proven prime; http://primes.utm.edu/primes/page.php?id=24043
29307576428proven prime; http://primes.utm.edu/primes/page.php?id=22126
30359997523proven prime; http://primes.utm.edu/primes/page.php?id=20235
31375117839proven prime; http://primes.utm.edu/primes/page.php?id=74907
325083310624proven prime; http://primes.utm.edu/primes/page.php?id=75849
338183917103proven prime; http://primes.utm.edu/primes/page.php?id=11084
3410491121925B. de Water, Apr. 2001proven prime; http://primes.utm.edu/primes/page.php?id=120463
3513002127173D. Fox, Dec. 2001
3614809130949T. D. Noe, Feb. 12, 2003
3720110742029H. Lifchitz, Feb. 2003
3839737983047H. Lifchitz, Aug. 2003
3943378190655H. Lifchitz, Sep. 2003
40590041123311H. Lifchitz, Jan. 2005
41593689124074H. Lifchitz, Jan. 2005
42604711126377H. Lifchitz, Feb. 2005
43931517194676H. Lifchitz, Sep. 2008
441049897219416H. Lifchitz, Oct. 2008
451285607268676H. Lifchitz, Nov. 2008
461636007341905H. Lifchitz, Mar. 2009
471803059376817H. Lifchitz, Jun. 2009
481968721411439H. Lifchitz, Nov. 2009
492904353606974H. Lifchitz, Jul. 2014
503244369678033H. Lifchitz, Sep. 2017

Here, F_(37511) was proven prime using the Coppersmith-Howgrave-Graham method (J. Renze, pers. comm., Aug. 16, 2005; Crandall and Pomerance 2005, p. 189), F_(50833) was proven prime by D. Broadhurst in Oct. 2005 using a CHG proof with ECPP helpers, and F_(81839) (Broadhurst 2001) and F_(104911) (in October 2015) have also been proven to be prime.

It is not known if there are an infinite number of Fibonacci primes.

See also

Fibonacci Number, Integer Sequence Primes, Lucas Prime, Prime Number, Probable Prime

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References

Brillhart, J.; Montgomery, P. L.; and Silverman, R. D. "Tables of Fibonacci and Lucas Factorizations." Math. Comput. 50, 251-260, 1988.Broadhurst, D. "Fibonacci(81839) is prime." 22 Apr 2001. http://listserv.nodak.edu/scripts/wa.exe?A2=ind0104&L=nmbrthry&P=R1807&D=0.Caldwell, C. "Fibonacci Number." http://primes.utm.edu/top20/page.php?id=39.Caldwell, C. "Fibonacci Prime." http://primes.utm.edu/glossary/page.php?sort=FibonacciPrime.Crandall, R. and Pomerance, C. Prime Numbers: A Computational Perspective, 2nd ed. New York: Springer-Verlag, 2005.Dubner, H. and Keller, W. "New Fibonacci and Lucas Primes." Math. Comput. 68, 417-427 and S1-S12, 1999.Gardner, M. Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American. New York: Knopf, 1979.Lifchitz, H. and Lifchitz, R. "PRP Top Records." http://www.primenumbers.net/prptop/searchform.php?form=F(n).Noe, T. D. and Vos Post, J. "Primes in Fibonacci n-step and Lucas n-step Sequences." J. Integer Seq. 8, Article 05.4.4., 2005.Pickover, C. A. Mazes for the Mind: Computers and the Unexpected. New York: St. Martin's Press, p. 350, 1993.Pickover, C. A. A Passion for Mathematics. New York: Wiley, p. 54, 2005.Ribenboim, P. The Little Book of Big Primes. New York: Springer-Verlag, p. 178, 1991.

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Fibonacci Prime

Cite this as:

Weisstein, Eric W. "Fibonacci Prime." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FibonacciPrime.html

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