Eisenstein primes
An Eisenstein integer is a non-unit Gaussian integer a + bω where ω(1+ω) = -1, and a and b are integers.
ω is generally chosen as a cube root of unity:
As a Gaussian integer, any Eisenstein integer is either a unit (an integer with a multiplicative inverse [±1, ±ω, ±(ω^-1)]), a prime (a number p such that if p divides xy, then p necessarily divides either x or y), or composite (a product of primes).
An Eisenstein integer a + bω is a prime if either it is a product of a unit and an integer prime p such that p % 3 == 2 or norm(a + bω) is an integer prime.
Eisenstein numbers can be generated by choosing any a and b such that a and b are integers. To allow generation in a relatively fixed order, such numbers can be ordered by their 2-norm or norm:
norm(eisenstein integer(a, b)) = |a + bω| =
- Task
- Find, and show here as their complex number values, the first 100 (by norm order) Eisenstein primes nearest 0.
- Stretch
- Plot in the complex plane at least the first 2000 such numbers (again, as found with norm closest to 0).
- See also
Julia
""" rosettacode.org/wiki/Eisenstein_primes """
import Base: Complex, real, imag
import LinearAlgebra: norm
import Primes: isprime
import Plots: scatter
struct Eisenstein{T<:Integer} <: Number
a::T
b::T
Eisenstein(a::T, b::T) where {T} = new{T}(a, b)
Eisenstein(a::T) where {T<:Real} = new{T}(a, zero(T))
Eisenstein(a::Real, b::Real) = new{eltype(promote(a, b))}(promote(a, b)...)
end
const ω = Eisenstein(false, true)
real(n::Eisenstein) = n.a - n.b / 2
imag(n::Eisenstein) = n.b * sqrt(big(3.0)) / 2
norm(n::Eisenstein) = n.a * n.a + n.b * n.b - n.a * n.b
Complex(n::Eisenstein{T}) where {T} = Complex{typeof(real(n))}(real(n), imag(n))
"""
is_eisenstein_prime(n)
An Eisenstein integer is a non-unit Gaussian integer a + bω where ω(1+ω) = -1,
and a and b are integers. As a Gaussian integer, any Eisenstein integer is
either a unit (an integer with a multiplicative inverse [±1, ±ω, ±(ω^-1)]),
prime (a number p such that if p divides xy, then p necessarily divides
either x or y), or composite (a product of primes).
An Eisenstein integer a + bω is a prime if either it is a product of a unit
and an integer prime p such that p % 3 == 2 or norm(a + bω) is an integer prime.
"""
function is_eisenstein_prime(n::Eisenstein)
if n.a == 0 || n.b == 0 || n.a === n.b
c = max(abs(n.a), abs(n.b))
return isprime(c) && c % 3 == 2
else
return isprime(norm(n))
end
end
function test_eisenstein_primes(graphlimitsquared = 10_000, printlimit = 100)
lim = isqrt(graphlimitsquared)
arr = [Eisenstein(a, b) for a = -lim:lim, b = -lim:lim]
eprimes = sort!([arr[p[1]] for p in enumerate(arr) if is_eisenstein_prime(p[2])],
lt = (x, y) -> norm(x) < norm(y))
for (i, c) in enumerate(eprimes)
if i <= printlimit
print(lpad(Complex{Float32}(Complex(c)), 24), i % 4 == 0 ? "\n" : "")
end
end
display(
scatter(
map(real, eprimes),
map(imag, eprimes),
markersize = 1,
title = "Eisenstein primes with norm < $lim",
),
)
end
test_eisenstein_primes()
- Output:
0.0f0 - 1.7320508f0im -1.5f0 - 0.8660254f0im 1.5f0 - 0.8660254f0im -1.5f0 + 0.8660254f0im 1.5f0 + 0.8660254f0im 0.0f0 + 1.7320508f0im -1.0f0 - 1.7320508f0im 1.0f0 - 1.7320508f0im -2.0f0 + 0.0f0im 2.0f0 + 0.0f0im -1.0f0 + 1.7320508f0im 1.0f0 + 1.7320508f0im -0.5f0 - 2.598076f0im 0.5f0 - 2.598076f0im -2.0f0 - 1.7320508f0im 2.0f0 - 1.7320508f0im -2.5f0 - 0.8660254f0im 2.5f0 - 0.8660254f0im -2.5f0 + 0.8660254f0im 2.5f0 + 0.8660254f0im -2.0f0 + 1.7320508f0im 2.0f0 + 1.7320508f0im -0.5f0 + 2.598076f0im 0.5f0 + 2.598076f0im -1.0f0 - 3.4641016f0im 1.0f0 - 3.4641016f0im -2.5f0 - 2.598076f0im 2.5f0 - 2.598076f0im -3.5f0 - 0.8660254f0im 3.5f0 - 0.8660254f0im -3.5f0 + 0.8660254f0im 3.5f0 + 0.8660254f0im -2.5f0 + 2.598076f0im 2.5f0 + 2.598076f0im -1.0f0 + 3.4641016f0im 1.0f0 + 3.4641016f0im -0.5f0 - 4.3301272f0im 0.5f0 - 4.3301272f0im -3.5f0 - 2.598076f0im 3.5f0 - 2.598076f0im -4.0f0 - 1.7320508f0im 4.0f0 - 1.7320508f0im -4.0f0 + 1.7320508f0im 4.0f0 + 1.7320508f0im -3.5f0 + 2.598076f0im 3.5f0 + 2.598076f0im -0.5f0 + 4.3301272f0im 0.5f0 + 4.3301272f0im -2.5f0 - 4.3301272f0im 2.5f0 - 4.3301272f0im -5.0f0 + 0.0f0im 5.0f0 + 0.0f0im -2.5f0 + 4.3301272f0im 2.5f0 + 4.3301272f0im -2.0f0 - 5.196152f0im 2.0f0 - 5.196152f0im -3.5f0 - 4.3301272f0im 3.5f0 - 4.3301272f0im -5.5f0 - 0.8660254f0im 5.5f0 - 0.8660254f0im -5.5f0 + 0.8660254f0im 5.5f0 + 0.8660254f0im -3.5f0 + 4.3301272f0im 3.5f0 + 4.3301272f0im -2.0f0 + 5.196152f0im 2.0f0 + 5.196152f0im -0.5f0 - 6.0621777f0im 0.5f0 - 6.0621777f0im -5.0f0 - 3.4641016f0im 5.0f0 - 3.4641016f0im -5.5f0 - 2.598076f0im 5.5f0 - 2.598076f0im -5.5f0 + 2.598076f0im 5.5f0 + 2.598076f0im -5.0f0 + 3.4641016f0im 5.0f0 + 3.4641016f0im -0.5f0 + 6.0621777f0im 0.5f0 + 6.0621777f0im -2.5f0 - 6.0621777f0im 2.5f0 - 6.0621777f0im -4.0f0 - 5.196152f0im 4.0f0 - 5.196152f0im -6.5f0 - 0.8660254f0im 6.5f0 - 0.8660254f0im -6.5f0 + 0.8660254f0im 6.5f0 + 0.8660254f0im -4.0f0 + 5.196152f0im 4.0f0 + 5.196152f0im -2.5f0 + 6.0621777f0im 2.5f0 + 6.0621777f0im -0.5f0 - 7.7942286f0im 0.5f0 - 7.7942286f0im -6.5f0 - 4.3301272f0im 6.5f0 - 4.3301272f0im -7.0f0 - 3.4641016f0im 7.0f0 - 3.4641016f0im -7.0f0 + 3.4641016f0im 7.0f0 + 3.4641016f0im -6.5f0 + 4.3301272f0im 6.5f0 + 4.3301272f0im