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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.

(up) Definitions and Notes

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).

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.

(up) Record Primes of this Type

rankprime digitswhowhencomment
1U(132409)/2882138154561602271737 27651 E16 Jul 2024 Fibonacci cofactor, ECPP
2U(120937)/241873 / 13689853218820385381 25250 E1 Apr 2024 Fibonacci cofactor, ECPP
3U(117167)/17658707237 24476 E1 Apr 2024 Fibonacci cofactor, ECPP
4U(106663)/35892566541651557 22275 E1 Mar 2024 Fibonacci cofactor, ECPP
5U(105509)/1441\
18801533126010445795676378394340544227572822879081
21997 E1 Jul 2022 Fibonacci cofactor, ECPP
6U(78919)/15574900936381642440917 16471 c77 Sep 2020 Fibonacci cofactor, ECPP
7U(71983)/5614673 / 363946049 15028 c77 Nov 2018 Fibonacci cofactor, ECPP
8U(69239)/1384781 14464 c77 Sep 2018 Fibonacci cofactor, ECPP
9U(66947)/12485272838388758877279873712131648167413 13951 c77 Apr 2017 Fibonacci cofactor, ECPP
10U(61813)/594517433 / 3761274442997 12897 c77 Dec 2018 Fibonacci cofactor, ECPP
11U(59369)/244242366914846603945\
83037561699885688092693836440759407570209763004757
12337 c79 Apr 2015 Fibonacci cofactor, ECPP
12U(54799)/4661437953906084533621577211561 11422 c8 Jan 2015 Fibonacci cofactor, ECPP
13U(54521)/6403194135342743624071073 11370 c8 Jan 2015 Fibonacci cofactor, ECPP
14U(53189)/69431662887136064191105625570683133711989 11075 c8 Dec 2014 Fibonacci cofactor, ECPP
15U(51031)/95846689435051369 10648 c8 Sep 2014 Fibonacci cofactor, ECPP
16U(43399)/4704006095\
75881344601538056264109423291827366228494341196421
9010 c8 Sep 2013 Fibonacci cofactor, ECPP
17U(42829)/107130175995197969243646842778153077 8916 c8 Apr 2014 Fibonacci cofactor, ECPP
18U(42043)/1681721 8780 c56 Feb 2012 Fibonacci cofactor, ECPP
19U(40763)/643247084652261620737 8498 c8 Nov 2013 Fibonacci cofactor, ECPP
20U(37987)/1832721858208455887947958246414213 7906 c39 Sep 2012 Fibonacci cofactor, ECPP

(up) 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 Fn and Ln for n>1000 whenever it is known.]
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