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(145353) 18689 c69 Dec 2013 ECPP, Lucas primitive part
3 primV(86756) 16920 c74 Jan 2015 Lucas primitive part, ECPP
4 primV(123573) 16198 c77 May 2019 Lucas primitive part, ECPP
5 primV(121227) 15890 c77 May 2019 Lucas primitive part, ECPP
6 primV(120258) 15649 c77 May 2019 Lucas primitive part, ECPP
7 primV(76568) 15034 c74 Oct 2015 Lucas primitive part, ECPP
8 primV(75316) 14897 c74 Dec 2015 Lucas primitive part, ECPP
9 primV(91322) 14847 c74 Feb 2016 Lucas primitive part, ECPP
10 primV(110676) 14713 c74 Jun 2016 Lucas primitive part, ECPP
11 primV(112914) 14446 c74 Nov 2016 Lucas primitive part, ECPP
12 primV(82630) 13814 c74 Sep 2014 Lucas primitive part, ECPP
13 primV(73549) 12324 c74 Jul 2015 Lucas primitive part, ECPP
14 primV(57724) 12063 p54 Jul 2001 Lucas primitive part, cyclotomy
15 primV(59018) 11789 c74 Jun 2015 Lucas primitive part, ECPP
16 primV(77231) 11637 c74 May 2015 Lucas primitive part, ECPP
17 primV(83481) 11631 c74 May 2015 Lucas primitive part, ECPP
18 primV(64652) 11577 c74 Apr 2015 Lucas primitive part, ECPP
19 primV(56356) 11557 c74 Feb 2015 Lucas primitive part, ECPP
20 primV(58672) 11557 c74 Feb 2015 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, 1995. New York, NY, 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