Lucas Aurifeuillian primitive part

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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) = F_2n(-ρ²)

ρ^f(2n)
,

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

L(2^rk) =
∏
d|k L^*(2^rd),

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(183835) 15368 c77 Mar 2019 Lucas Aurifeuillian primitive part, ECPP
2 primB(181705) 15189 c77 Feb 2019 Lucas Aurifeuillian primitive part, ECPP
3 primB(268665) 14972 c77 Apr 2019 Lucas Aurifeuillian primitive part, ECPP
4 primA(284895) 14626 c77 Apr 2019 Lucas Aurifeuillian primitive part, ECPP
5 primA(170575) 14258 c77 Dec 2018 Lucas Aurifeuillian primitive part, ECPP
6 primB(163595) 13675 c77 Jun 2018 Lucas Aurifeuillian primitive part, ECPP
7 primB(242295) 13014 c77 Jun 2018 Lucas Aurifeuillian primitive part, ECPP
8 primA(154415) 12728 c77 Jun 2018 Lucas Aurifeuillian primitive part, ECPP
9 primA(263865) 12570 c77 Jun 2018 Lucas Aurifeuillian primitive part, ECPP
10 primA(143705) 11703 c77 Apr 2017 Lucas Aurifeuillian primitive part, ECPP
11 primB(219165) 11557 c77 May 2015 Lucas Aurifeuillian primitive part, ECPP
12 primA(219135) 10462 c8 Sep 2014 Lucas Aurifeuillian primitive part, ECPP
13 primA(196035) 9359 c8 May 2014 Lucas Aurifeuillian primitive part, ECPP
14 primA(159165) 8803 c8 Nov 2013 Lucas Aurifeuillian primitive part, ECPP
15 primB(148605) 8282 c8 Oct 2013 Lucas Aurifeuillian primitive part, ECPP
16 primB(103645) 8202 c8 Oct 2013 Lucas Aurifeuillian primitive part, ECPP
17 primB(119945) 8165 c8 Sep 2013 Lucas Aurifeuillian primitive part, ECPP
18 primB(99835) 8126 c8 Sep 2013 Lucas Aurifeuillian primitive part, ECPP
19 primB(96545) 8070 c8 Sep 2013 Lucas Aurifeuillian primitive part, ECPP
20 primB(145545) 7824 c8 Sep 2013 Lucas Aurifeuillian primitive part, ECPP

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