

A005478


Prime Fibonacci numbers.
(Formerly M0741)


71



2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917, 475420437734698220747368027166749382927701417016557193662268716376935476241
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OFFSET

1,1


COMMENTS

a(n) == 1 (mod 4) for n > 2. (Proof. Otherwise 3 < a(n) = F_k == 3 (mod 4). Then k == 4 (mod 6) (see A079343 and A161553) and so k is not prime. But k is prime since F_k is prime and k != 4  see Caldwell.)
More generally, A190949(n) == 1 (mod 4).  N. J. A. Sloane
With the exception of 3, every term of this sequence has a prime index in the sequence of Fibonacci numbers (A000045); e.g., 5 is the fifth Fibonacci number, 13 is the seventh Fibonacci number, 89 the eleventh, etc.  Alonso del Arte, Aug 16 2013
Note: A001605 gives those indices.  Antti Karttunen, Aug 16 2013
The six known safe primes 2p + 1 such that p is a Fibonacci prime are in A263880; the values of p are in A155011. There are only two known Fibonacci primes p for which 2p  1 is also prime, namely, p = 2 and 3. Is there a reason for this bias toward prime 2p + 1 over 2p  1 among Fibonacci primes p?  Jonathan Sondow, Nov 04 2015


REFERENCES

J. Brillhart, P. L. Montgomery and R. D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), 251260.
J.M. De Koninck, Ces nombres qui nous fascinent, Entry 89, p. 32, Ellipses, Paris 2008.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..23
C. Caldwell's The Top Twenty, Fibonacci Number.
Ron Knott, Mathematics of the Fibonacci Series
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci nstep and Lucas nstep Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4
Eric Weisstein's World of Mathematics, Fibonacci Prime


FORMULA

a(n) = A000045(A001605(n)). A000040 INTERSECT A000045.  R. J. Mathar, Nov 01 2007


MATHEMATICA

Select[Fibonacci[Range[400]], PrimeQ] (* Alonso del Arte, Oct 13 2011 *)


PROG

(PARI) je=[]; for(n=0, 400, if(isprime(fibonacci(n)), je=concat(je, fibonacci(n)))); je
(Sage) [i for i in fibonacci_xrange(0, 10^80) if is_prime(i)] # Bruno Berselli, Jun 26 2014


CROSSREFS

Cf. A001605, A000045, A030426, A075736, A263880.
Subsequence of A178762.
Column k=1 of A303216.
Sequence in context: A139589 A152114 A139095 * A117740 A041047 A120494
Adjacent sequences: A005475 A005476 A005477 * A005479 A005480 A005481


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

Sequence corrected by Enoch Haga, Feb 11 2000
One more term from Jason Earls, Jul 12 2001
Comment and proof added by Jonathan Sondow, May 24 2011


STATUS

approved



