The Top Twenty--a Prime Page Collection

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.

(up) 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.

(up) Record Primes of this Type

rankprime digitswhowhencomment
1primA(82975) 6935 p54 Jul 2001 Lucas Aurifeuillian primitive part
2primA(53155) 4444 x25 Jun 2002 Lucas Aurifeuillian primitive part, cyclotomy
3primA(52825) 4414 x25 May 2003 Lucas Aurifeuillian primitive part
4primA(42685) 3568 c46 Jul 2008 Lucas Aurifeuillian primitive part
5primA(41665) 3211 c8 Jul 2003 Lucas Aurifeuillian primitive part, ECPP
6primA(51945) 2894 c8 Jun 2003 Lucas Aurifeuillian primitive part, ECPP
7primB(49785) 2774 c8 Jun 2003 Lucas Aurifeuillian primitive part, ECPP
8primA(52275) 2676 c8 Jun 2003 Lucas Aurifeuillian primitive part, ECPP
9primB(48375) 2634 F3 Jun 2001 Lucas Aurifeuillian primitive part, APR-CL assisted
10primB(31145) 2603 p54 May 2001 Lucas Aurifeuillian primitive part
11primB(34045) 2584 c8 May 2003 Lucas Aurifeuillian primitive part, ECPP
12primA(47235) 2538 c8 May 2003 Lucas Aurifeuillian primitive part, ECPP
13primB(53625) 2508 c8 Jun 2003 Lucas Aurifeuillian primitive part, ECPP
14primB(45105) 2407 c8 May 2003 Lucas Aurifeuillian primitive part, ECPP
15primA(50715) 2318 c8 May 2003 Lucas Aurifeuillian primitive part, ECPP
16primA(26875) 2195 F3 Jun 2001 Lucas Aurifeuillian primitive part, APR-CL assisted
17primB(26155) 2186 p54 May 2001 Lucas Aurifeuillian primitive part
18primA(25945) 2169 F3 Jun 2001 Lucas Aurifeuillian primitive part, APR-CL assisted
19primB(29995) 2148 c8 May 2003 Lucas Aurifeuillian primitive part, ECPP
20primB(27625) 2007 c8 Jun 2001 Lucas Aurifeuillian primitive part, ECPP

(up) Related Pages

(up) 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
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