%I M2309 N0911 #176 Oct 30 2024 10:28:02

%S 3,4,5,7,11,13,17,23,29,43,47,83,131,137,359,431,433,449,509,569,571,

%T 2971,4723,5387,9311,9677,14431,25561,30757,35999,37511,50833,81839,

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

%N Indices of prime Fibonacci numbers.

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

%C 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

%C 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_

%C _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

%C Terms n such that n-2 is also a term are listed in A279795. - _M. F. Hasler_, Dec 24 2016

%C 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, there are no other Fibonacci twin primes. - _Bobby Jacobs_, Sep 23 2017

%C 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

%D Clifford A. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 350.

%D Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 54.

%D Paulo Ribenboim, The Little Book of Big Primes, Springer-Verlag, NY, 1991, p. 178.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Paul Bourdelais, <a href="/A001605/b001605.txt">Table of n, a(n) for n = 1..56</a> (first 51 terms from Henri Lifchitz)

%H P. Bourdelais, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906">A Generalized Repunit Conjecture</a>

%H J. Brillhart, P. L. Montgomery and R. D. Silverman, <a href="http://dx.doi.org/10.1090/S0025-5718-1988-0917832-6">Tables of Fibonacci and Lucas factorizations</a>, Math. Comp. 50 (1988), 251-260.

%H David Broadhurst, <a href="http://groups.yahoo.com/group/primeform/files/LucasFib/">Fibonacci Numbers</a>

%H David Broadhurst, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;a1468dd4.0104">Proof that F(81839) is prime</a>, NMBRTHRY Mailing List, 22 April 2001

%H Chris K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=FibonacciPrime">Fibonacci prime</a>

%H Rosina Campbell, Duc Van Huynh, Tyler Melton, and Andrew Percival, <a href="https://arxiv.org/abs/1710.05687">Elliptic Curves of Fibonacci order over F_p</a>, arXiv:1710.05687 [math.NT], 2017.

%H H. Dubner and W. Keller, <a href="http://dx.doi.org/10.1090/S0025-5718-99-00981-3">New Fibonacci and Lucas Primes</a>, Math. Comp. 68 (1999) 417-427.

%H Dudley Fox, <a href="http://www.gristle.to/markup/primes/ppfibs.html">Search for Possible Fibonacci Primes</a>

%H Dov Jarden, <a href="/A001602/a001602.pdf">Recurring Sequences</a>, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 36.

%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html">Mathematics of the Fibonacci Series</a>

%H Alex Kontorovich and Jeff Lagarias, <a href="https://arxiv.org/abs/1808.03235">On Toric Orbits in the Affine Sieve</a>, arXiv:1808.03235 [math.NT], 2018.

%H Henri & Renaud Lifchitz, <a href="http://www.primenumbers.net/prptop/searchform.php?form=F%28n%29">PRP Records.</a>

%H Tony D. Noe and Jonathan Vos Post, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Noe/noe5.html">Primes in Fibonacci n-step and Lucas n-step Sequences,</a> J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

%H R. Ondrejka, <a href="http://www.utm.edu/research/primes/lists/top_ten/">The Top Ten: a Catalogue of Primal Configurations</a>

%H PRP Top Records, <a href="http://www.primenumbers.net/prptop/searchform.php?form=F%28n%29&amp;action=Search">Search for: F(n)</a>

%H Lawrence Somer and Michal Křížek, <a href="https://www.fq.math.ca/Papers1/53-1/SomerKrizek5222014.pdf">On Primes in Lucas Sequences</a>, Fibonacci Quart. 53 (2015), no. 1, 2-23.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciPrime.html">Fibonacci Prime</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IntegerSequencePrimes.html">Integer Sequence Primes</a>

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

%t Select[Range[10^4], PrimeQ[Fibonacci[#]] &] (* _Harvey P. Dale_, Nov 20 2012 *)

%t (* Start ~ 1.8x faster than the above *)

%t Select[Range[10^4], PrimeQ[#] && PrimeQ[Fibonacci[#]] &] (* _Eric W. Weisstein_, Nov 07 2017 *)

%t Select[Prime[Range[PrimePi[10^4]]], PrimeQ[Fibonacci[#]] &] (* _Eric W. Weisstein_, Nov 07 2017 *)

%t (* End *)

%o (PARI) v=[3,4]; forprime(p=5,1e5, if(ispseudoprime(fibonacci(p)), v=concat(v,p))); v \\ _Charles R Greathouse IV_, Feb 14 2011

%o (PARI) is_A001605(n)={n==4 || isprime(n) & ispseudoprime(fibonacci(n))} \\ _M. F. Hasler_, Sep 29 2012

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

%Y Subsequence of A046022.

%Y Column k=1 of A303215.

%K nonn,hard,nice,changed

%O 1,1

%A _N. J. A. Sloane_

%E Additional comments from _Robert G. Wilson v_, Aug 18 2000

%E More terms from _David Broadhurst_, Nov 08 2001

%E Two more terms (148091 and 201107) from _T. D. Noe_, Feb 12 2003 and Mar 04 2003

%E 397379 from _T. D. Noe_, Aug 18 2003

%E 433781, 590041, 593689 from _Henri Lifchitz_ submitted by _Ray Chandler_, Feb 11 2005

%E 604711 from _Henri Lifchitz_ communicated by _Eric W. Weisstein_, Nov 29 2005

%E 931517, 1049897, 1285607 found by _Henri Lifchitz_ circa Nov 01 2008 and submitted by _Alexander Adamchuk_, Nov 28 2008

%E 1636007 from _Henri Lifchitz_ March 2009, communicated by _Eric W. Weisstein_, Apr 24 2009

%E 1803059 and 1968721 from _Henri Lifchitz_, November 2009, submitted by _Alex Ratushnyak_, Aug 08 2012

%E a(49)=2904353 from _Henri Lifchitz_, Jul 15 2014

%E a(50)=3244369 from _Henri Lifchitz_, Nov 04 2017

%E a(51)=3340367 from _Henri Lifchitz_, Apr 25 2018

%E a(52)-a(56) from _Ryan Propper_ added by _Paul Bourdelais_, Aug 23 2024