Fibonacci Prime
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.
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 PrimeExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
Fibonacci PrimeCite this as:
Weisstein, Eric W. "Fibonacci Prime." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FibonacciPrime.html