Lucas 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 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(86477)/1042112515940998434071039 18049 c77 Sep 2020 Lucas cofactor, ECPP
2 V(80761)/(23259169 · 24510801979) 16861 c77 Aug 2020 Lucas cofactor, ECPP
3 V(77417)/313991497376559420151 16159 c77 Aug 2020 Lucas cofactor, ECPP
4 V(68213)/7290202116115634431 14237 c77 Aug 2018 Lucas cofactor, ECPP
5 V(66533)/2128184670585621839884209100279 13875 c77 Dec 2018 Lucas cofactor, ECPP
6 V(52859)/1124137922466041911 11029 c8 Oct 2014 Lucas cofactor, ECPP
7 V(52201)/705\
85804042896975505694709575919458733851279868446609 10857 c8 Mar 2015 Lucas cofactor, ECPP
8 V(52009)/39772636393178951550299332730909 10838 c8 Mar 2015 Lucas cofactor, ECPP
9 V(51941)/28080521576109021\
14547210696868337380250300924116591143641642866931 10789 c8 Jan 2015 Lucas cofactor, ECPP
10 V(51349)/224417260052884218046541 10708 c8 Sep 2014 Lucas cofactor, ECPP
11 V(50989)/69818796119453411 10640 c8 Aug 2014 Lucas cofactor, ECPP
12 V(49391)/298414424560419239 10305 c8 Jun 2013 Lucas cofactor, ECPP
13 V(45953)/4561241750239 9591 c56 Mar 2012 Lucas cofactor, ECPP
14 V(43987)/175949 9188 c8 May 2014 Lucas cofactor, ECPP
15 V(39769)/18139109172816581 8295 c8 Nov 2013 Lucas cofactor, ECPP
16 V(39607)/158429 8273 c46 Dec 2011 Lucas cofactor, ECPP
17 V(37357)/20210113386303842894568629 7782 c8 Sep 2013 Lucas cofactor, ECPP
18 V(35107)/525110138418084707309 7317 c8 Sep 2013 Lucas cofactor, ECPP
19 V(34759)/27112021 7257 c33 Jul 2005 Lucas cofactor, ECPP
20 V(33353)/279902102741094707003083072429 6941 c8 May 2013 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.]