Migration complete - please let us know if anything isn't working.
Fibonacci Number
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.
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
un+1 = un + un-1 (n > 2).
It is easy to show that un divides unm (see primitive part of a Fibonacci number), so for un 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 un 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(148091) 30949 x49 Sep 2021 Fibonacci number, ECPP 2 U(130021) 27173 x48 May 2021 Fibonacci number, ECPP 3 U(104911) 21925 c82 Oct 2015 Fibonacci number, ECPP 4 U(81839) 17103 p54 Apr 2001 Fibonacci number 5 U(50833) 10624 CH4 Oct 2005 Fibonacci number 6 U(37511) 7839 x13 Jun 2005 Fibonacci number 7 U(35999) 7523 p54 Jul 2001 Fibonacci number, cyclotomy 8 U(30757) 6428 p54 Jul 2001 Fibonacci number, cyclotomy 9 U(25561) 5342 p54 Jul 2001 Fibonacci number 10 U(14431) 3016 p54 Apr 2001 Fibonacci number 11 U(9677) 2023 c2 Nov 2000 Fibonacci number, ECPP 12 U(9311) 1946 DK Mar 1995 Fibonacci number 13 U(5387) 1126 WM Jan 1991 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
Printed from the PrimePages <t5k.org> © Reginald McLean.