A001605 Indices of prime Fibonacci numbers.

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, 104911, 130021, 148091, 201107, 397379, 433781, 590041, 593689, 604711, 931517, 1049897, 1285607, 1636007, 1803059, 1968721, 2904353, 3244369, 3340367

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• nice

Comments

Some of the larger entries may only correspond to probable primes.

Since F(n) divides F(mn) (cf. A001578, A086597), all terms of this sequence are primes except for a(2) = 4 = 2 * 2 but F(2) = 1. - M. F. Hasler, Dec 12 2007

What is the next larger twin prime after F(4) = 3, F(5) = 5, F(7) = 13? The next candidates seem to be F(104911) or F(1968721) (greater of a pair), or F(397379), F(931517) (lesser of a pair). - M. F. Hasler, Jan 30 2013, edited Dec 24 2016, edited Sep 23 2017 by Bobby Jacobs

_Henri Lifchitz_ confirms that the data section gives the full list (49 terms) as far as we know it today of indices of prime Fibonacci numbers (including proven primes and PRPs). - N. J. A. Sloane, Jul 09 2016

Terms n such that n-2 is also a term are listed in A279795. - M. F. Hasler, Dec 24 2016

There are no Fibonacci numbers that are twin primes after F(7) = 13. Every Fibonacci prime greater than F(4) = 3 is of the form F(2*n+1). Since F(2*n+1)+2 and F(2*n+1)-2 are F(n+2)*L(n-1) and F(n-1)*L(n+2) in some order, and F(n+2) > 1, L(n-1) > 1, F(n-1) > 1, and L(n+2) > 1 for n > 3. - Bobby Jacobs, Sep 23 2017

These primes are occurring with about the same normalized frequency as Repunit primes (see Generalized Repunit Conjecture Ref). Assuming a base=1.618 (ratio of sequential terms), then the best fit coefficient is 0.60324 for the first 56 terms, which is already approaching Euler's constant 0.56145948. - Paul Bourdelais, Aug 23 2024

References

• Clifford A. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 350.
• Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 54.
• Paulo Ribenboim, The Little Book of Big Primes, Springer-Verlag, NY, 1991, p. 178.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Paul Bourdelais, Table of n, a(n) for n = 1..56 (first 51 terms from Henri Lifchitz)
• Paul Bourdelais, A Generalized Repunit Conjecture
• John Brillhart, Peter L. Montgomery and Robert D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), 251-260.
• David Broadhurst, Fibonacci Numbers
• David Broadhurst, Proof that F(81839) is prime, NMBRTHRY Mailing List, Apr 22 2001.
• Chris K. Caldwell, The Prime Glossary, Fibonacci prime
• Rosina Campbell, Duc Van Huynh, Tyler Melton, and Andrew Percival, Elliptic Curves of Fibonacci order over F_p, arXiv:1710.05687 [math.NT], 2017.
• Harvey Dubner and Wilfrid Keller, New Fibonacci and Lucas Primes, Math. Comp. 68 (1999) 417-427.
• Dudley Fox, Search for Possible Fibonacci Primes
• Shyam Sunder Gupta, Fabulous Fibonacci Numbers, Lucas Numbers, and Golden Ratio, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 8, 223-274.
• Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 36.
• Ron Knott, Mathematics of the Fibonacci Series
• Alex Kontorovich and Jeff Lagarias, On Toric Orbits in the Affine Sieve, arXiv:1808.03235 [math.NT], 2018.
• Henri & Renaud Lifchitz, PRP Records.
• Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4.
• Rudolf Ondrejka, The Top Ten: a Catalogue of Primal Configurations
• PRP Top Records, Search for: F(n)
• Lawrence Somer and Michal Křížek, On Primes in Lucas Sequences, Fibonacci Quart. 53 (2015), no. 1, 2-23.
• Eric Weisstein's World of Mathematics, Fibonacci Prime
• Eric Weisstein's World of Mathematics, Integer Sequence Primes

Crossrefs

Cf. A000045, A001578, A005478, A080345, A086597, A117595.

Subsequence of A046022.

Column k=1 of A303215.

Programs

Mathematica

Select[Range[10^4], PrimeQ[Fibonacci[#]] &] (* Harvey P. Dale, Nov 20 2012 *)
(* Start ~ 1.8x faster than the above *)
Select[Range[10^4], PrimeQ[#] && PrimeQ[Fibonacci[#]] &] (* Eric W. Weisstein, Nov 07 2017 *)
Select[Prime[Range[PrimePi[10^4]]], PrimeQ[Fibonacci[#]] &] (* Eric W. Weisstein, Nov 07 2017 *)
(* End *)

PARI

v=[3,4]; forprime(p=5,1e5, if(ispseudoprime(fibonacci(p)), v=concat(v,p))); v \\ Charles R Greathouse IV, Feb 14 2011

PARI

is_A001605(n)={n==4 || isprime(n) & ispseudoprime(fibonacci(n))}  \\ M. F. Hasler, Sep 29 2012

Formula

Prime(i) = a(n) for some n <=> A080345(i) <= 1. - M. F. Hasler, Dec 12 2007

Extensions

Additional comments from Robert G. Wilson v, Aug 18 2000

More terms from David Broadhurst, Nov 08 2001

Two more terms (148091 and 201107) from T. D. Noe, Feb 12 2003 and Mar 04 2003

397379 from T. D. Noe, Aug 18 2003

433781, 590041, 593689 from Henri Lifchitz submitted by Ray Chandler, Feb 11 2005

604711 from Henri Lifchitz communicated by Eric W. Weisstein, Nov 29 2005

931517, 1049897, 1285607 found by Henri Lifchitz circa Nov 01 2008 and submitted by Alexander Adamchuk, Nov 28 2008

1636007 from Henri Lifchitz March 2009, communicated by Eric W. Weisstein, Apr 24 2009

1803059 and 1968721 from Henri Lifchitz, November 2009, submitted by Alex Ratushnyak, Aug 08 2012

a(49)=2904353 from Henri Lifchitz, Jul 15 2014

a(50)=3244369 from Henri Lifchitz, Nov 04 2017

a(51)=3340367 from Henri Lifchitz, Apr 25 2018

a(52)-a(56) from Ryan Propper added by Paul Bourdelais, Aug 23 2024