Lucas 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 Lucas numbers can be defined as follows:

v₁ = 1, v₂ = 3 and v_n+1 = v_n + v_n-1 (n > 2)

It can be shown that, for odd m, v_n divides v_nm. 

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

Record Primes of this Type

rank prime digits who when comment 1 V(277273)/11090921 57940 E4 May 2026 Lucas cofactor, ECPP 2 V(145703)/179214691 30442 E4 Nov 2023 Lucas cofactor, ECPP 3 V(145193)/38621339 30336 E4 Nov 2023 Lucas cofactor, ECPP 4 V(137483)/297756567358709 28718 E1 May 2026 Lucas cofactor, ECPP 5 V(132647)/30508811 / 445072057883695169 27697 E1 May 2026 Lucas cofactor, ECPP 6 V(122869)/40546771 / 1243743094029841 25656 E1 Apr 2024 Lucas cofactor, ECPP 7 V(116593)/120790349 24359 E4 Nov 2023 Lucas cofactor, ECPP 8 V(100417)/71304290377910160751180879\ 90532064354948544338847967474370719436805470448849 20911 E1 Apr 2024 Lucas cofactor, ECPP 9 V(98081)/31189759 / 611955609270431 / 6902594225498651 / 641303018340927841 20442 E1 Apr 2024 Lucas cofactor, ECPP 10 V(91943)/551659 / 2390519 / 9687119153094919 19187 E1 Jul 2022 Lucas cofactor, ECPP 11 V(91873)/3674921 / 193484539 / 167745030829 19175 E1 Jul 2022 Lucas cofactor, ECPP 12 V(86477)/1042112515940998434071039 18049 c77 Sep 2020 Lucas cofactor, ECPP 13 V(80761)/570100885555095451 16861 c77 Aug 2020 Lucas cofactor, ECPP 14 V(77417)/313991497376559420151 16159 c77 Aug 2020 Lucas cofactor, ECPP 15 V(68213)/7290202116115634431 14237 c77 Aug 2018 Lucas cofactor, ECPP 16 V(66533)/2128184670585621839884209100279 13875 c77 Dec 2018 Lucas cofactor, ECPP 17 V(64063)/464426465381142115542697818362662865912299 13347 E1 Aug 2024 Lucas cofactor, ECPP 18 V(52859)/1124137922466041911 11029 c8 Oct 2014 Lucas cofactor, ECPP 19 V(52201)/705\ 85804042896975505694709575919458733851279868446609 10857 c8 Mar 2015 Lucas cofactor, ECPP 20 V(52009)/39772636393178951550299332730909 10838 c8 Mar 2015 Lucas 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.]