Lucas Aurifeuillian 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

David Broadhurst writes: The Lucas numbers are defined by L(n) = L(n-1)+L(n-2), with L(0) = 2 and L(1) = 1. It follows that

L(n) = ρⁿ + (-ρ)^-n,

where ρ = (1+√5)/2 is the golden ratio. They are related to the Fibonacci numbers

F(n) =

ρn-(-ρ)-n √5

by L(n)=F(2n)/F(n), for n ± 0. The primitive part of L(n) is

L*(n) =

F2n(-ρ2) ρf(2n)

,

for n > 1. With L^*(1) = 1, the factorization

L(2rk) =

∏ d|k

L*(2rd),

results, for r ± 0 and odd k.

When n=5k, with odd k, there is also an Aurifeuillian factorization

L(5k) = L(k)A(5k)B(5k),
A(5k) = 5F(k)(F(k)-1)+1,
B(5k) = 5F(k)(F(k)+1)+1.

The Lucas Aurifeuillian primitive parts of L^*(n) = A^*(n)B^*(n) are

A^*(n) = gcd(L^*(n),A(n)), B^*(n) = gcd (L^*(n),B(n)),

for n=5 (mod 10). They may be computed in terms of the Möbius transformations

A±(n) =

∏ d|n d2 = ±1 (mod 5)

A(n/d)m(d), B±(n) =

∏ d|n d2 = ±1 (mod 5)

B(n/d)m(d),

which are not, in general, integers. The integer-valued primitive parts are

A^*(n) = A⁺(n)B^-(n), B^*(n) = B⁺(n)A^-(n),

with n = 5 (mod 10).

A^*(n) is prime for n = 25, 35, 45, 55, 65, 75, 85, 95, 105, 125, 145, 165, 185, 275, 335, 355, 535, 655, 735, 805, 925, 955, 1095, 1195, 1215, 1275, 1305, 1325, 1435, 1575, 1655, 1765, 2015, 2205, 2715, 2745, 2885, 3905, 3935, 4275, 5705, 5995, 7755, 8565, and for no other n < 10⁴.

B^*(n) is prime for n = 5, 15, 25, 35, 45, 75, 85, 105, 145, 155, 165, 185, 225, 255, 305, 315, 325, 335, 355, 365, 375, 475, 485, 525, 565, 575, 635, 695, 715, 765, 885, 1235, 1325, 1375, 1515, 2255, 2285, 3085, 3185, 3355, 3565, 3745, 3885, 4325, 4995, 5525, 5915, 6195, 6565, 6975, 6995, 7785, 8855, 9435, 9925, and for no other n < 10⁴.

A^*(n) and B^*(n) are simultaneously prime for n = 25, 35, 45, 75, 85, 105, 145, 165, 185, 335, 355, 1325, and for no other n < 10⁵.

Record Primes of this Type

rank prime digits who when comment 1 primB(282035) 21758 E1 Oct 2023 Lucas Aurifeuillian primitive part, ECPP 2 primA(296695) 21137 E1 Oct 2023 Lucas Aurifeuillian primitive part, ECPP 3 primA(413205) 21127 E1 Sep 2023 Lucas Aurifeuillian primitive part, ECPP 4 primB(220895) 18465 E1 Jun 2022 Lucas Aurifeuillian primitive part, ECPP 5 primB(235015) 17856 E1 May 2022 Lucas Aurifeuillian primitive part, ECPP 6 primA(201485) 16535 E1 May 2022 Lucas Aurifeuillian primitive part, ECPP 7 primB(225785) 16176 E1 May 2022 Lucas Aurifeuillian primitive part, ECPP 8 primB(183835) 15368 c77 Mar 2019 Lucas Aurifeuillian primitive part, ECPP 9 primB(181705) 15189 c77 Feb 2019 Lucas Aurifeuillian primitive part, ECPP 10 primB(268665) 14972 c77 Apr 2019 Lucas Aurifeuillian primitive part, ECPP 11 primA(284895) 14626 c77 Apr 2019 Lucas Aurifeuillian primitive part, ECPP 12 primA(170575) 14258 c77 Dec 2018 Lucas Aurifeuillian primitive part, ECPP 13 primB(163595) 13675 c77 Jun 2018 Lucas Aurifeuillian primitive part, ECPP 14 primB(242295) 13014 c77 Jun 2018 Lucas Aurifeuillian primitive part, ECPP 15 primA(154415) 12728 c77 Jun 2018 Lucas Aurifeuillian primitive part, ECPP 16 primA(263865) 12570 c77 Jun 2018 Lucas Aurifeuillian primitive part, ECPP 17 primA(143705) 11703 c77 Apr 2017 Lucas Aurifeuillian primitive part, ECPP 18 primB(219165) 11557 c77 May 2015 Lucas Aurifeuillian primitive part, ECPP 19 primA(219135) 10462 c8 Sep 2014 Lucas Aurifeuillian primitive part, ECPP 20 primA(196035) 9359 c8 May 2014 Lucas Aurifeuillian primitive part, ECPP

Related Pages

• Lucas factorizations 2 ≤ n < 10000

References

BMS88

J. Brillhart, P. L. Montgomery and R. D. Silverman, "Tables of Fibonacci and Lucas factorizations," Math. Comp., 50 (1988) 251--260, S1--S15.  MR 89h:11002 [See also [DK99].]

DK99

H. Dubner and W. Keller, "New Fibonacci and Lucas primes," Math. Comp., 68:225 (1999) 417--427, S1--S12.  MR 99c:11008 [Probable primality of F, L, F* and L* tested for n up to 50000, 50000, 20000, and 15000, respectively. Many new primes and algebraic factorizations found.]

Schinzel62

A. Schinzel, "On primitive prime factors of aⁿ - bⁿ," Proc. Cambridge Phil. Soc., 58 (1962) 555--562.  MR 26:1280

Stevenhagen87

P. Stevenhagen, "On Aurifeuillian factorizations," Nederl. Akad. Wetensch. Indag. Math., 49:4 (1987) 451--468.  MR 89a:11015