Fibonacci cofactor

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The Prime Pages

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

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).

In the same way we defined a Mersenne cofactor, we define a Fibonacci cofactor to be a proper divisor of a Fibonacci number with a prime index.

Record Primes of this Type

rank prime digits who when comment

1 U(30671)/1141737296775689 6395 c41 Dec 2005 Fibonacci cofactor, ECPP
2 U(21577)/(8626362776257 · 608114436652075009) 4479 c8 May 2004 Fibonacci cofactor, ECPP
3 U(19433)/(8200903423639793 · 124790158973035710313 · 163702910239586286961573) 4002 c8 May 2004 Fibonacci cofactor, ECPP
4 U(19051)/44198321 3974 c8 Apr 2004 Fibonacci cofactor, ECPP
5 U(18919)/1497228584233 3942 c8 Apr 2004 Fibonacci cofactor, ECPP
6 U(18427)/(1828363793 · 23130933997 · 11364458229549793) 3815 c8 Apr 2004 Fibonacci cofactor, ECPP
7 U(17137)/328335144266897 3567 c8 Mar 2004 Fibonacci cofactor, ECPP
8 U(17011)/42109783293497 3542 c8 Mar 2004 Fibonacci cofactor
9 U(16369)/(299022540829 · 8523458822385578881) 3391 c8 Mar 2004 Fibonacci cofactor, ECPP
10 U(14177)/2199986861 2954 c8 Mar 2004 Fibonacci cofactor
11 U(13063)/(4389169 · 10370895133369) 2710 c8 Mar 2004 Fibonacci cofactor
12 U(12743)/8869129 2656 c8 Mar 2004 Fibonacci cofactor
13 U(11807)/(231936709 · 1129990105451996281) 2441 c8 May 2004 Fibonacci cofactor, ECPP
14 U(11393)/(38772745844002993 · 2788032223609253801) 2346 c8 May 2004 Fibonacci cofactor, ECPP
15 U(11279)/(34987457 · 70263726073) 2339 c8 Mar 2004 Fibonacci cofactor
16 U(11093)/544296421641613 2304 c8 Mar 2004 Fibonacci cofactor
17 U(9931)/(3079452003211541 · 122039293401017981 · 604956242930486632441) 2022 c8 Mar 2004 Fibonacci cofactor
18 U(9967)/(120801834061 · 3105570884441 · 1047583984031437 · 22727240300133999677 · 15086016388211003643481) 2003 c8 Feb 2005 Fibonacci cofactor, ECPP
19 U(8537)/(900766408671673 · 558828605201 · 17191134589117 · 9229577309895207361) 1725 c8 Mar 2004 Fibonacci cofactor
20 U(8353)/(176465477 · 2184683363573 · 3620673479519515599362549) 1701 c8 Mar 2004 Fibonacci cofactor

References

BMS1988
J. Brillhart, P. Montgomery and R. Silverman, "Tables of Fibonacci and Lucas factorizations," Math. Comp., 50 (1988) 251--260. MR 89h:11002
DK1999
H. Dubner and W. Keller, "New fibonacci and lucas primes," Math. Comp., 68 (1999) 417--427. MR 99c:11008 [A supplement gives the complete factorization of F_n and L_n for n>1000 whenever it is known.]