# 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.

### 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

rankprime digitswhowhencomment
1primA(275285) 23012 E1 Mar 2024 Lucas Aurifeuillian primitive part, ECPP
2primB(282035) 21758 E1 Oct 2023 Lucas Aurifeuillian primitive part, ECPP
3primA(276335) 21736 E1 Mar 2024 Lucas Aurifeuillian primitive part, ECPP
4primA(296695) 21137 E1 Oct 2023 Lucas Aurifeuillian primitive part, ECPP
5primA(413205) 21127 E1 Sep 2023 Lucas Aurifeuillian primitive part, ECPP
6primB(220895) 18465 E1 Jun 2022 Lucas Aurifeuillian primitive part, ECPP
7primB(235015) 17856 E1 May 2022 Lucas Aurifeuillian primitive part, ECPP
8primA(201485) 16535 E1 May 2022 Lucas Aurifeuillian primitive part, ECPP
9primB(225785) 16176 E1 May 2022 Lucas Aurifeuillian primitive part, ECPP
10primB(183835) 15368 c77 Mar 2019 Lucas Aurifeuillian primitive part, ECPP
11primB(181705) 15189 c77 Feb 2019 Lucas Aurifeuillian primitive part, ECPP
12primB(268665) 14972 c77 Apr 2019 Lucas Aurifeuillian primitive part, ECPP
13primA(284895) 14626 c77 Apr 2019 Lucas Aurifeuillian primitive part, ECPP
14primA(170575) 14258 c77 Dec 2018 Lucas Aurifeuillian primitive part, ECPP
15primB(163595) 13675 c77 Jun 2018 Lucas Aurifeuillian primitive part, ECPP
16primB(242295) 13014 c77 Jun 2018 Lucas Aurifeuillian primitive part, ECPP
17primA(154415) 12728 c77 Jun 2018 Lucas Aurifeuillian primitive part, ECPP
18primA(263865) 12570 c77 Jun 2018 Lucas Aurifeuillian primitive part, ECPP
19primA(143705) 11703 c77 Apr 2017 Lucas Aurifeuillian primitive part, ECPP
20primB(219165) 11557 c77 May 2015 Lucas Aurifeuillian primitive part, ECPP

### 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