# Lucas Aurifeuillian primitive part

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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

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
1primB(163595) 13675 c77 Jun 2018 Lucas Aurifeuillian primitive part, ECPP
2primB(242295) 13014 c77 Jun 2018 Lucas Aurifeuillian primitive part, ECPP
3primA(154415) 12728 c77 Jun 2018 Lucas Aurifeuillian primitive part, ECPP
4primA(263865) 12570 c77 Jun 2018 Lucas Aurifeuillian primitive part, ECPP
5primA(143705) 11703 c77 Apr 2017 Lucas Aurifeuillian primitive part, ECPP
6primB(219165) 11557 c77 May 2015 Lucas Aurifeuillian primitive part, ECPP
7primA(219135) 10462 c8 Sep 2014 Lucas Aurifeuillian primitive part, ECPP
8primA(196035) 9359 c8 May 2014 Lucas Aurifeuillian primitive part, ECPP
9primA(159165) 8803 c8 Nov 2013 Lucas Aurifeuillian primitive part, ECPP
10primB(148605) 8282 c8 Oct 2013 Lucas Aurifeuillian primitive part, ECPP
11primB(103645) 8202 c8 Oct 2013 Lucas Aurifeuillian primitive part, ECPP
12primB(119945) 8165 c8 Sep 2013 Lucas Aurifeuillian primitive part, ECPP
13primB(99835) 8126 c8 Sep 2013 Lucas Aurifeuillian primitive part, ECPP
14primB(96545) 8070 c8 Sep 2013 Lucas Aurifeuillian primitive part, ECPP
15primB(145545) 7824 c8 Sep 2013 Lucas Aurifeuillian primitive part, ECPP
16primA(161595) 7313 c8 Sep 2013 Lucas Aurifeuillian primitive part, ECPP
17primB(134415) 7163 c8 May 2013 Lucas Aurifeuillian primitive part, ECPP
18primA(82975) 6935 p54 Jul 2001 Lucas Aurifeuillian primitive part
19primA(123405) 6502 c8 Apr 2013 Lucas Aurifeuillian primitive part, ECPP
20primA(118275) 6170 c8 Mar 2013 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