A005478 - OEIS (original) (raw)

A005478

Prime Fibonacci numbers.
(Formerly M0741)

89

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

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.)

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

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.-M. De Koninck, Ces nombres qui nous fascinent, Entry 89, p. 32, Ellipses, Paris 2008.

R. K. Guy, Unsolved Problems in Number Theory, Section A3.

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

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

(Python)

from itertools import islice

from sympy import isprime

def A005478_gen(): # generator of terms

a, b = 1, 1

while True:

if isprime(b):

yield b

a, b = b, a+b

EXTENSIONS

Sequence corrected by Enoch Haga, Feb 11 2000