Fibonacci cofactor

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

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(105509)/1441\ 18801533126010445795676378394340544227572822879081 21997 E1 Jul 2022 Fibonacci cofactor, ECPP 2 U(78919)/15574900936381642440917 16471 c77 Sep 2020 Fibonacci cofactor, ECPP 3 U(71983)/5614673 / 363946049 15028 c77 Nov 2018 Fibonacci cofactor, ECPP 4 U(69239)/1384781 14464 c77 Sep 2018 Fibonacci cofactor, ECPP 5 U(66947)/12485272838388758877279873712131648167413 13951 c77 Apr 2017 Fibonacci cofactor, ECPP 6 U(61813)/594517433 / 3761274442997 12897 c77 Dec 2018 Fibonacci cofactor, ECPP 7 U(59369)/244242366914846603945\ 83037561699885688092693836440759407570209763004757 12337 c79 Apr 2015 Fibonacci cofactor, ECPP 8 U(54799)/4661437953906084533621577211561 11422 c8 Jan 2015 Fibonacci cofactor, ECPP 9 U(54521)/6403194135342743624071073 11370 c8 Jan 2015 Fibonacci cofactor, ECPP 10 U(53189)/69431662887136064191105625570683133711989 11075 c8 Dec 2014 Fibonacci cofactor, ECPP 11 U(51031)/95846689435051369 10648 c8 Sep 2014 Fibonacci cofactor, ECPP 12 U(43399)/4704006095\ 75881344601538056264109423291827366228494341196421 9010 c8 Sep 2013 Fibonacci cofactor, ECPP 13 U(42829)/107130175995197969243646842778153077 8916 c8 Apr 2014 Fibonacci cofactor, ECPP 14 U(42043)/1681721 8780 c56 Feb 2012 Fibonacci cofactor, ECPP 15 U(40763)/643247084652261620737 8498 c8 Nov 2013 Fibonacci cofactor, ECPP 16 U(37987)/(16117960073 · 94533840409 · 1202815961509) 7906 c39 Sep 2012 Fibonacci cofactor, ECPP 17 U(37217)/4466041 7771 c46 Dec 2011 Fibonacci cofactor, ECPP 18 U(34897)/4599458691503517435329 7272 c8 Aug 2013 Fibonacci cofactor, ECPP 19 U(34807)/551750980997908879677508732866536453 7239 c8 Jul 2013 Fibonacci cofactor, ECPP 20 U(34607)/13088506284255296513 7213 c8 Jul 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.]