150914(2)

Ruby


xx + 27yy 型の素数

tsujimotter さんの本日のブログ

(http://tsujimotter.hatenablog.com/entry/xx-27yy)

について、自分で実験してみることにした。

require 'prime'

# m次以下を取り出す
def mul(f_ary, b_ary, m)
  s1, s2 = f_ary.size, b_ary.size
  ary = Array.new(s1 + s2 - 1, 0)
  (0..s1 - 1).each{|i|
    (0..s2 - 1).each{|j|
      ary[i + j] += f_ary[i] * b_ary[j]
    }
  }
  ary[0..m]
end

# 1 - q^(6n)
def f1(n)
  f1 = Array.new(6 * n + 1, 0)
  f1[0] = 1
  f1[-1] = -1
  f1
end

# 1 - q^(18n)
def f2(n)
  f2 = Array.new(18 * n + 1, 0)
  f2[0] = 1
  f2[-1] = -1
  f2
end

ary = [0, 1]
n = 1
while 6 * n <= 300
  ary = mul(ary, f1(n), 300)
  n += 1
end
n = 1
while 18 * n <= 300
  ary = mul(ary, f2(n), 300)
  n += 1
end
p ary

Prime.each(300){|i| p [i, ary[i]]}

出力結果

[0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

[2, 0]

[3, 0]

[5, 0]

[7, -1]

[11, 0]

[13, -1]

[17, 0]

[19, -1]

[23, 0]

[29, 0]

[31, 2]

[37, -1]

[41, 0]

[43, 2]

[47, 0]

[53, 0]

[59, 0]

[61, -1]

[67, -1]

[71, 0]

[73, -1]

[79, -1]

[83, 0]

[89, 0]

[97, -1]

[101, 0]

[103, -1]

[107, 0]

[109, 2]

[113, 0]

[127, 2]

[131, 0]

[137, 0]

[139, -1]

[149, 0]

[151, -1]

[157, 2]

[163, -1]

[167, 0]

[173, 0]

[179, 0]

[181, -1]

[191, 0]

[193, -1]

[197, 0]

[199, -1]

[211, -1]

[223, 2]

[227, 0]

[229, 2]

[233, 0]

[239, 0]

[241, -1]

[251, 0]

[257, 0]

[263, 0]

[269, 0]

[271, -1]

[277, 2]

[281, 0]

[283, 2]

[293, 0]

赤字の素数がxx + 27yy 型の素数となる。