Lucas primitive part

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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

Ribenboim's book (pp. 54--83) gives an excellent review. Generalized Lucas numbers were introduced in [Lucas1878] and intensively studied in [Carmichael1913]. Their role in primality proving was cemented by [Morrison75]. Their primitive parts (also known as Sylvester's cyclotomic numbers) were studied in [Ward1959]. Prime generalized Lucas numbers are clearly a particular case of prime primitive parts, occurring when n is also a prime. As Ribenboim indicates, there is an extensive literature on primitive prime Lucas factors, from [Carmichael1913] to [Voutier1995], via, for example, [Schinzel1974] and [Stewart1977].

S(n)=primV(2,-1,n)

Record Primes of this Type

rank prime digits who when comment

1 primV(205011) 28552 x39 May 2009 Lucas primitive part
2 primV(57724) 12063 p54 Jul 2001 Lucas primitive part, cyclotomy
3 primV(77058) 10729 CH3 Sep 2005 Lucas primitive part
4 primV(77841) 10496 x25 Sep 2005 Lucas primitive part
5 primV(39124) 8176 CH3 Sep 2005 Lucas primitive part
6 primV(48381) 6741 x23 Dec 2005 Lucas primitive part
7 primV(39700) 6621 p54 Apr 2001 Lucas primitive part
8 primV(29657) 6057 c11 Nov 2009 Lucas primitive part, ECPP
9 primV(28844) 6028 p12 Apr 2001 Lucas primitive part
10 primV(25504) 5324 F3 May 2001 Lucas primitive part, APR-CL assisted
11 primV(24998) 5033 c48 Nov 2008 Lucas primitive part, ECPP
12 primV(24383) 4951 c4 Sep 2009 Lucas primitive part, ECPP
13 primV(27167) 4866 c4 Sep 2009 Lucas primitive part, ECPP
14 primV(28940) 4836 c4 Jul 2009 Lucas primitive part, ECPP
15 primV(29810) 4515 c8 Jul 2004 Lucas primitive part, ECPP
16 primV(20578) 4301 p54 May 2001 Lucas primitive part
17 primV(21298) 4249 c4 Apr 2009 Lucas primitive part, ECPP
18 primV(19148) 4001 p12 Apr 2001 Lucas primitive part
19 primV(18857) 3882 c4 Mar 2009 Lucas primitive part, ECPP
20 primV(23716) 3863 p54 Apr 2001 Lucas primitive part

References

Carmichael1913
R. D. Carmichael, "On the numerical factors of the arithmetic forms αⁿ ± βⁿ," Ann. Math., 15 (1913) 30--70.
Lucas1878
E. Lucas, "Theorie des fonctions numeriques simplement periodiques," Amer. J. Math., 1 (1878) 184--240 and 289--231.
Morrison75
M. Morrison, "A note on primality testing using Lucas sequences," Math. Comp., 29 (1975) 181--182. MR 51:5469
Ribenboim95
P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995. pp. xxiv+541, ISBN 0-387-94457-5. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]
Schinzel1974
A. Schinzel, "Primitive divisors of the expression Aⁿ - Bⁿ in algebraic number fields," J. Reine Angew. Math., 268/269 (1974) 27--33. MR 49:8961
Stewart1977
C. L. Stewart, "On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers," Proc. Lond. Math. Soc., 35:3 (1977) 425--447. MR 58:10694
Voutier1995
Voutier, P. M., "Primitive divisors of Lucas and Lehmer sequences," Math. Comp., 64:210 (1995) 869--888. MR1284673 (Annotation available)
Ward1959
M. Ward, "Tests for primality based on Sylvester's cyclotomic numbers," Pacific J. Math., 9 (1959) 1269--1272. MR 21:7180