A Fibonacci prime is a Fibonacci number that is also a prime number. Every that is prime must have a prime index , with the exception of . However, the converse is not true (i.e., not every prime index gives a prime ).
The first few (possibly probable) prime Fibonacci numbers are 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ... (OEIS A005478), corresponding to indices , 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, ... (OEIS A001605). (Note that Gardner's statement that is prime (Gardner 1979, p. 161) is incorrect, especially since 531 is not even prime, which it must be for to be prime.) The following table summarizes Fibonacci (possibly probable) primes with index .
term | index | digits | discoverer | status |
24 | 5387 | 1126 | proven prime; http://primes.utm.edu/primes/page.php?id=51129 | |
25 | 9311 | 1946 | proven prime; http://primes.utm.edu/primes/page.php?id=37470 | |
26 | 9677 | 2023 | proven prime; http://primes.utm.edu/primes/page.php?id=35537 | |
27 | 14431 | 3016 | proven prime; http://primes.utm.edu/primes/page.php?id=29537 | |
28 | 25561 | 5342 | proven prime; http://primes.utm.edu/primes/page.php?id=24043 | |
29 | 30757 | 6428 | proven prime; http://primes.utm.edu/primes/page.php?id=22126 | |
30 | 35999 | 7523 | proven prime; http://primes.utm.edu/primes/page.php?id=20235 | |
31 | 37511 | 7839 | proven prime; http://primes.utm.edu/primes/page.php?id=74907 | |
32 | 50833 | 10624 | proven prime; http://primes.utm.edu/primes/page.php?id=75849 | |
33 | 81839 | 17103 | proven prime; http://primes.utm.edu/primes/page.php?id=11084 | |
34 | 104911 | 21925 | B. de Water, Apr. 2001 | proven prime; http://primes.utm.edu/primes/page.php?id=120463 |
35 | 130021 | 27173 | D. Fox, Dec. 2001 | |
36 | 148091 | 30949 | T. D. Noe, Feb. 12, 2003 | |
37 | 201107 | 42029 | H. Lifchitz, Feb. 2003 | |
38 | 397379 | 83047 | H. Lifchitz, Aug. 2003 | |
39 | 433781 | 90655 | H. Lifchitz, Sep. 2003 | |
40 | 590041 | 123311 | H. Lifchitz, Jan. 2005 | |
41 | 593689 | 124074 | H. Lifchitz, Jan. 2005 | |
42 | 604711 | 126377 | H. Lifchitz, Feb. 2005 | |
43 | 931517 | 194676 | H. Lifchitz, Sep. 2008 | |
44 | 1049897 | 219416 | H. Lifchitz, Oct. 2008 | |
45 | 1285607 | 268676 | H. Lifchitz, Nov. 2008 | |
46 | 1636007 | 341905 | H. Lifchitz, Mar. 2009 | |
47 | 1803059 | 376817 | H. Lifchitz, Jun. 2009 | |
48 | 1968721 | 411439 | H. Lifchitz, Nov. 2009 | |
49 | 2904353 | 606974 | H. Lifchitz, Jul. 2014 | |
50 | 3244369 | 678033 | H. Lifchitz, Sep. 2017 |
Here, was proven prime using the Coppersmith-Howgrave-Graham method (J. Renze, pers. comm., Aug. 16, 2005; Crandall and Pomerance 2005, p. 189), was proven prime by D. Broadhurst in Oct. 2005 using a CHG proof with ECPP helpers, and (Broadhurst 2001) and (in October 2015) have also been proven to be prime.
It is not known if there are an infinite number of Fibonacci primes.