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(71983)/5614673 / 363946049 15028 c77 Nov 2018 Fibonacci cofactor, ECPP
2 U(69239)/1384781 14464 c77 Sep 2018 Fibonacci cofactor, ECPP
3 U(66947)/12485272838388758877279873712131648167413 13951 c77 Apr 2017 Fibonacci cofactor, ECPP
4 U(61813)/594517433 / 3761274442997 12897 c77 Dec 2018 Fibonacci cofactor, ECPP
5 U(59369)/244242366914846603945\
83037561699885688092693836440759407570209763004757 12337 c79 Apr 2015 Fibonacci cofactor, ECPP
6 U(54799)/4661437953906084533621577211561 11422 c8 Jan 2015 Fibonacci cofactor, ECPP
7 U(54521)/6403194135342743624071073 11370 c8 Jan 2015 Fibonacci cofactor, ECPP
8 U(53189)/69431662887136064191105625570683133711989 11075 c8 Dec 2014 Fibonacci cofactor, ECPP
9 U(51031)/95846689435051369 10648 c8 Sep 2014 Fibonacci cofactor, ECPP
10 U(43399)/4704006095\
75881344601538056264109423291827366228494341196421 9010 c8 Sep 2013 Fibonacci cofactor, ECPP
11 U(42829)/107130175995197969243646842778153077 8916 c8 Apr 2014 Fibonacci cofactor, ECPP
12 U(42043)/1681721 8780 c56 Feb 2012 Fibonacci cofactor, ECPP
13 U(40763)/643247084652261620737 8498 c8 Nov 2013 Fibonacci cofactor, ECPP
14 U(37987)/(16117960073 · 94533840409 · 1202815961509) 7906 c39 Sep 2012 Fibonacci cofactor, ECPP
15 U(37217)/4466041 7771 c46 Dec 2011 Fibonacci cofactor, ECPP
16 U(34897)/4599458691503517435329 7272 c8 Aug 2013 Fibonacci cofactor, ECPP
17 U(34807)/551750980997908879677508732866536453 7239 c8 Jul 2013 Fibonacci cofactor, ECPP
18 U(34607)/13088506284255296513 7213 c8 Jul 2013 Fibonacci cofactor, ECPP
19 U(33997)/81\
19544695419968014626314520991088099382355441843013 7053 c8 Apr 2013 Fibonacci cofactor, ECPP
20 U(32077)/153087505413829037510511957221947361 6669 c8 May 2013 Fibonacci cofactor, ECPP

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.]