A279795 -id:A279795 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

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

Cf. A000045, A001578, A005478, A080345, A086597, A117595.
Subsequence of A046022.
Column k=1 of A303215.

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 *)

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

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

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

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

A281087 Numbers k such that Fibonacci(k) and Fibonacci(k+2) are both prime.

Original entry on oeis.org

3, 5, 11, 431, 569

Offset: 1

Bobby Jacobs, Jan 14 2017

Smaller primes of the Fibonacci prime pairs in A073340.

See the comment to A073340 - Harvey P. Dale, Jan 30 2025

			11 is in the sequence because Fibonacci(11) = 89 and Fibonacci(13) = 233 are both prime.

J. B. Cosgrave and K. Dilcher, Pairs of reciprocal quadratic congruences involving primes, Fig. Quart. 51 (2) (2013) 98-111

First differs from A101315 at a(5).

Cf. A000045, A001359, A001605, A073340, A101315, A279795.

Select[Range[600],PrimeQ[Fibonacci[#]] && PrimeQ[Fibonacci[#+2]] &] (* Stefano Spezia, Nov 15 2024 *)
SequencePosition[Table[If[PrimeQ[Fibonacci[n]],1,0],{n,600}],{1,,1}][[;;,1]] (* _Harvey P. Dale, Jan 30 2025 *)

a(n) = A279795(n) - 2.

a(n) = A073340(2n-1).

A073340 Fibonacci prime pairs: the indices of each pair differ by two and the relevant Fibonacci numbers are both prime.

Original entry on oeis.org

3, 5, 5, 7, 11, 13, 431, 433, 569, 571

Offset: 1

Harvey P. Dale, Aug 25 2002

There are no other Fibonacci prime pairs up to Fibonacci(104911). (See A001605.) Are there any larger terms?

			The 431st Fibonacci number and the 433rd Fibonacci number are both prime and their indices differ by 2.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Rev. ed. 1997, p. 46.

Cf. A000045, A001605, A279795, A281087.

Flatten[Select[Partition[Select[Range[3000], PrimeQ[Fibonacci[ # ]]&], 2, 1], #[[2]] - #[[1]] == 2 &]]

from sympy import isprime
def afind(limit):
  i, fnm2, fnm1 = 1, 1, 1
  while i < limit:
    if isprime(fnm2) and isprime(fnm2 + fnm1):
      print(i, i+2, sep=", ", end=", ")
    i, fnm2, fnm1 = i+1, fnm1, fnm2 + fnm1
afind(600) # Michael S. Branicky, Mar 05 2021

Offset changed to 1 by Joerg Arndt, Jan 18 2017

a(1) and a(2) prepended by Bobby Jacobs, Jan 18 2017

A297624 Numbers k such that Fibonacci(2k+1) and Fibonacci(2k-1) are prime.

Original entry on oeis.org

2, 3, 6, 216, 285

Offset: 1

Vincenzo Librandi, Jan 08 2018

a(6) > 1622184 if it exists (see A001605). - Chai Wah Wu, Jan 23 2018

			2 is in the sequence because F(3)=2 and F(5)=5 are prime.
6 is in the sequence because F(11)=89 and F(13)=233 are prime.

Cf. A000045, A001605, A073340, A117517, A117595, A279795, A281087.

o := [];; for k in [1..500] do if IsPrime(Fibonacci(2*k+1)) and IsPrime(Fibonacci(2*k-1)) then Add(o,k); fi; od; A297624 := o; # Muniru A Asiru, Jan 25 2018

[n: n in [0..700] | IsPrime(Fibonacci(2*n+1)) and  IsPrime(Fibonacci(2*n-1))];

with(combinat, fibonacci): select(k -> isprime(fibonacci(2*k+1)) and isprime(fibonacci(2*k-1)), [$1..500]); # Muniru A Asiru, Jan 25 2018

Select[Range[0, 3000], PrimeQ[Fibonacci[2 # + 1]] && PrimeQ[Fibonacci[2 # - 1]] &]

isok(n) = isprime(fibonacci(2*n-1)) && isprime(fibonacci(2*n+1)); \\ Michel Marcus, Jan 08 2018

from sympy import isprime
A297624_list, k, a, b, c, aflag = [], 1, 1, 1, 2, False
while k < 1000:
    cflag = isprime(c)
    if aflag and cflag:
        A297624_list.append(k)
    k, a, b, c, aflag = k + 1, c, b + c, b + 2*c, cflag # Chai Wah Wu, Jan 23 2018

From Chai Wah Wu, Jan 23 2018: (Start)
a(n) = (A279795(n)-1)/2 = (A281087(n)+1)/2 = (A073340(2n-1)+1)/2.
For n > 1, a(n) == 0 mod 3 as otherwise Fibonacci(2*k+1) or Fibonacci(2*k-1) is even. (End)

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A001605 Indices of prime Fibonacci numbers.

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A281087 Numbers k such that Fibonacci(k) and Fibonacci(k+2) are both prime.

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A073340 Fibonacci prime pairs: the indices of each pair differ by two and the relevant Fibonacci numbers are both prime.

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A297624 Numbers k such that Fibonacci(2*k+1) and Fibonacci(2*k-1) are prime.

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A297624 Numbers k such that Fibonacci(2k+1) and Fibonacci(2k-1) are prime.