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Fibonacci Number |
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]
>rank prime digits who when comment 1 U(104911) 21925 c82 Oct 2015 Fibonacci number, ECPP 2 U(81839) 17103 p54 Apr 2001 Fibonacci number 3 U(50833) 10624 CH4 Oct 2005 Fibonacci number 4 U(37511) 7839 x13 Jun 2005 Fibonacci number 5 U(35999) 7523 p54 Jul 2001 Fibonacci number, cyclotomy 6 U(30757) 6428 p54 Jul 2001 Fibonacci number, cyclotomy 7 U(25561) 5342 p54 Jul 2001 Fibonacci number 8 U(14431) 3016 p54 Apr 2001 Fibonacci number 9 U(9677) 2023 c2 Nov 2000 Fibonacci number, ECPP 10 U(9311) 1946 DK Mar 1995 Fibonacci number 11 U(5387) 1126 WM Dec 1990 Fibonacci number
- 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