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(34759)/27112021 7257 c33 Jul 2005 Lucas cofactor, ECPP
2 V(22811)/(2469062641 · 84961206854418761) 4741 c8 Jun 2004 Lucas cofactor, ECPP
3 V(17029)/(9570299 · 495749440031) 3541 c8 Mar 2004 Lucas cofactor, ECPP
4 V(15511)/394599841 3234 c8 Mar 2004 Lucas cofactor, ECPP
5 V(14887)/1256071867381 3100 c8 Mar 2004 Lucas cofactor
6 V(12503)/4954888889 2604 c8 Mar 2004 Lucas cofactor
7 V(12457)/(1420099 · 81953391325049801) 2581 c8 May 2004 Lucas cofactor, ECPP
8 V(11657)/69172639 2429 c8 Mar 2004 Lucas cofactor
9 V(11393)/(3076111 · 5299498382701) 2362 c8 Mar 2004 Lucas cofactor
10 V(11261)/16823009787209 2341 c8 Mar 2004 Lucas cofactor
11 V(11213)/(224261 · 1476324252027331181) 2320 c8 May 2004 Lucas cofactor, ECPP
12 V(11173)/(25206289 · 28583254701767411959 · 24018475125955094159731) 2286 c8 May 2004 Lucas cofactor, ECPP
13 V(10223)/61893855542632111 2120 c8 May 2004 Lucas cofactor, ECPP
14 V(9839)/491951 2051 c8 Mar 2004 Lucas cofactor
15 V(9209)/(29745071 · 101409509 · 547683904686691 · 4025749704474499) 1879 c8 Mar 2004 Lucas cofactor
16 V(8807)/356419291 1833 c8 Mar 2004 Lucas cofactor
17 V(8329)/(5533286033716829 · 62798306322672929543921 · 2398413145658705436562211) 1678 c8 Jun 2006 Lucas cofactor, ECPP
18 V(7243)/289721 1509 c8 Mar 2004 Lucas cofactor
19 V(6379)/(9823661 · 187797761 · 39735824399) 1308 c8 Mar 2004 Lucas cofactor
20 V(6661)/(20413264084399 · 254844997471 · 6472729219639 · 8036635984600095627961 · 64250170013936618067104660874142964379889) 1292 c8 Feb 2005 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.]