Lucas primitive part

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

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(194181) 24908 E1 Oct 2024 Lucas primitive part, ECPP
3 primV(119162) 24903 E1 Oct 2024 Lucas primitive part, ECPP
4 primV(214470) 23895 E1 Oct 2024 Lucas primitive part, ECPP
5 primV(143234) 23654 E1 Oct 2024 Lucas primitive part, ECPP
6 primV(110723) 22997 E1 Oct 2024 Lucas primitive part, ECPP
7 primV(180906) 22905 E1 Sep 2024 Lucas primitive part, ECPP
8 primV(154281) 21495 E4 Nov 2023 Lucas primitive part, ECPP
9 primV(112028) 20063 E1 Jun 2022 Lucas primitive part, ECPP
10 primV(151521) 19863 E1 Jun 2022 Lucas primitive part, ECPP
11 primV(145353) 18689 c69 Dec 2013 ECPP, Lucas primitive part
12 primV(153279) 18283 E1 May 2022 Lucas primitive part, ECPP
13 primV(148197) 17696 E1 May 2022 Lucas primitive part, ECPP
14 primV(169830) 17335 E1 May 2022 Lucas primitive part, ECPP
15 primV(101510) 16970 E1 May 2022 Lucas primitive part, ECPP
16 primV(86756) 16920 c74 Jan 2015 Lucas primitive part, ECPP
17 primV(122754) 16653 c77 Feb 2021 Lucas primitive part, ECPP
18 primV(123573) 16198 c77 May 2019 Lucas primitive part, ECPP
19 primV(121227) 15890 c77 May 2019 Lucas primitive part, ECPP
20 primV(120258) 15649 c77 May 2019 Lucas primitive part, ECPP

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