Fibonacci Number

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The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page. This page is about one of those forms. Comments and suggestions requested.

Definitions and Notes

A Fibonacci prime, as you should easily guess, is a Fibonacci number that is prime. Recall that the Fibonacci numbers can be defined as follows: u₁ = u₂ = 1 and u_n+1 = u_n + u_n-1 (n > 2).

It is easy to show that u_n divides u_nm (see primitive part of a Fibonacci number), so for u_n to be a prime, the subscript must either be 4 (because u₂=1) or a prime. This however is not sufficient!

The list of known Fibonacci primes begins u_n with

n = 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 and 81839.

They are probable-prime for n = 104911 [Bouk de Water], 130021 [D. Fox], 148091 [T. D. Noe] and 201107, 397379, 433781 [H. Lifchitz]

Record Primes of this Type

rank prime digits who when comment

1 U(130021) 27173 x48 May 2021 Fibonacci number, ECPP
2 U(104911) 21925 c82 Oct 2015 Fibonacci number, ECPP
3 U(81839) 17103 p54 Apr 2001 Fibonacci number
4 U(50833) 10624 CH4 Oct 2005 Fibonacci number
5 U(37511) 7839 x13 Jun 2005 Fibonacci number
6 U(35999) 7523 p54 Jul 2001 Fibonacci number, cyclotomy
7 U(30757) 6428 p54 Jul 2001 Fibonacci number, cyclotomy
8 U(25561) 5342 p54 Jul 2001 Fibonacci number
9 U(14431) 3016 p54 Apr 2001 Fibonacci number
10 U(9677) 2023 c2 Nov 2000 Fibonacci number, ECPP
11 U(9311) 1946 DK Mar 1995 Fibonacci number
12 U(5387) 1126 WM Dec 1990 Fibonacci number

References

BMS1988
J. Brillhart, P. Montgomery and R. Silverman, "Tables of Fibonacci and Lucas factorizations," Math. Comp., 50 (1988) 251--260. MR 89h:11002
Brillhart1999
J. Brillhart, "Note on Fibonacci primality testing," Fibonacci Quart., 36:3 (1998) 222--228. MR1627388
DK99
H. Dubner and W. Keller, "New Fibonacci and Lucas primes," Math. Comp., 68:225 (1999) 417--427, S1--S12. MR 99c:11008 [Probable primality of F, L, F* and L* tested for n up to 50000, 50000, 20000, and 15000, respectively. Many new primes and algebraic factorizations found.]
LRS1999
Leyendekkers, J. V., Rybak, J. M. and Shannon, A. G., "An analysis of Mersenne-Fibonacci and Mersenne-Lucas primes," Notes Number Theory Discrete Math., 5:1 (1999) 1--26. MR 1738744