The Top Twenty: Lucas Aurifeuillian primitive part

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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.

rank

prime

digits

who

when

comment

primB(183835)

15368

c77

Mar 2019