160515(4)

Ruby


Two refinements of Lagrange's four-square theorem

せきゅーんさんのブログに紹介があった
以下のプレプリントに興味を持った。
https://arxiv.org/pdf/1605.03074v2.pdf

任意の自然数n に対し、
ある整数x, y, z, w が存在し、
n = x^2 + y^2 + z^2 + w^2,
x + y + z + w は平方数
とできることを証明している。

例えば、
160515 = 37249 + 39601 + 40401 + 43264 = 193^2 + (-199)^2 + (-201)^2 + 208^2,
193 - 199 - 201 + 208 = 1 = 1^2
とできる。

各n に対し、このようなx, y, z, w を見つけてみた。

def square?(n)
  return false if n < 0
  Math.sqrt(n).to_i ** 2 == n
end

def f(ary)
  x, y, z, w = ary
  # マイナス符号のつけかたを考えるのは最小限にとどめる。
  [[ x,  y,  z,  w],
   [ x,  y,  z, -w],
   [ x,  y, -z,  w],
   [ x,  y, -z, -w],
   [ x, -y,  z,  w],
   [ x, -y,  z, -w],
   [ x, -y, -z,  w],
   [ x, -y, -z, -w],
   [-x,  y,  z,  w],
   [-x,  y,  z, -w],
   [-x,  y, -z,  w],
   [-x,  y, -z, -w],
   [-x, -y,  z,  w],
   [-x, -y,  z, -w],
   [-x, -y, -z,  w],
   [-x, -y, -z, -w]].each{|a|
    return [1, a] if square?(a.inject(:+))
  }
  [0, ary]
end

def g(n)
  ary = []
  (0..Math.sqrt(n).to_i).each{|x|
    (x..Math.sqrt(n - x * x).to_i).each{|y|
      (y..Math.sqrt(n - x * x - y * y).to_i).each{|z|
        w2 = n - x * x - y * y - z * z
        if w2 >= z * z && square?(w2)
          a = f([x, y, z, Math.sqrt(w2).to_i])
          ary << a[1] if a[0] == 1
        end
      }
    }
  }
  ary
end

(1..200).each{|i| p [i, g(i)]}

出力結果
[1, [[0, 0, 0, 1]]]
[2, [[0, 0, 1, -1]]]
[3, [[0, 1, 1, -1]]]
[4, [[1, 1, 1, 1]]]
[5, [[0, 0, -1, 2]]]
[6, [[0, 1, 1, 2]]]
[7, [[1, 1, 1, -2]]]
[8, [[0, 0, 2, 2]]]
[9, [[0, 1, 2, -2]]]
[10, [[0, 0, 1, 3], [1, -1, 2, 2]]]
[11, [[0, -1, -1, 3]]]
[12, [[1, 1, 1, -3]]]
[13, [[0, 0, -2, 3], [-1, 2, 2, -2]]]
[14, [[0, 1, 2, -3]]]
[15, [[1, 1, 2, -3]]]
[16, [[0, 0, 0, 4], [2, 2, 2, -2]]]
[17, [[0, 2, 2, -3]]]
[18, [[0, 0, 3, -3], [0, 1, -1, 4], [1, 2, -2, 3]]]
[19, [[0, 1, 3, -3], [-1, -1, -1, 4]]]
[20, [[1, -1, 3, -3]]]
[21, [[0, -1, -2, 4], [2, 2, 2, 3]]]
[22, [[0, -2, 3, 3], [1, 1, 2, -4]]]
[23, [[1, 2, 3, 3]]]
[24, [[0, 2, 2, -4]]]
[25, [[0, 0, -3, 4], [1, 2, 2, 4]]]
[26, [[0, 0, -1, 5], [0, 1, 3, -4], [2, 2, 3, -3]]]
[27, [[0, 3, 3, 3], [1, 1, 3, 4]]]
[28, [[1, -1, -1, 5], [1, 3, 3, -3]]]
[29, [[0, 2, 3, 4]]]
[30, [[0, 1, -2, 5], [1, 2, -3, 4]]]
[31, [[1, 1, 2, 5], [-2, 3, 3, -3]]]
[32, [[0, 0, 4, -4]]]
[33, [[0, 1, 4, 4], [0, 2, 2, 5], [2, -2, -3, 4]]]
[34, [[0, 3, -3, 4], [1, -1, 4, -4], [1, 2, 2, -5]]]
[35, [[0, 1, 3, 5], [-1, 3, 3, 4]]]
[36, [[1, 1, 3, -5], [3, 3, -3, -3]]]
[37, [[-1, 2, 4, 4], [2, 2, 2, -5]]]
[38, [[0, -1, -1, 6], [0, 2, 3, -5], [2, 3, 3, -4]]]
[39, [[1, 1, 1, 6], [1, 2, 3, -5]]]
[40, [[0, 0, -2, 6], [2, 2, 4, -4]]]
[41, [[0, 0, 4, 5], [0, 1, 2, 6]]]
[42, [[0, 1, 4, -5], [1, -1, -2, 6], [1, 3, 4, -4], [-2, -2, 3, 5]]]
[43, [[0, 3, 3, -5], [1, 1, 4, -5], [3, -3, -3, 4]]]
[44, [[1, -3, -3, 5]]]
[45, [[0, 0, 3, 6], [0, 2, 4, -5], [-1, 2, 2, 6], [-2, 3, 4, 4]]]
[46, [[0, 1, -3, 6], [1, 2, -4, 5]]]
[47, [[1, -1, 3, 6], [2, -3, -3, 5]]]
[48, [[0, 4, 4, -4], [2, 2, 2, -6]]]
[49, [[0, -2, -3, 6], [2, -2, 4, 5]]]
[50, [[0, 0, 5, -5], [0, 3, -4, 5], [1, 2, 3, -6], [3, -3, 4, -4]]]
[51, [[0, 1, 1, 7], [0, 1, 5, -5], [-1, 3, 4, -5]]]
[52, [[-1, -1, -1, 7], [1, -1, 5, -5], [3, 3, 3, -5]]]
[53, [[0, 0, 2, 7], [0, -1, 4, 6], [2, 2, 3, -6]]]
[54, [[0, -1, -2, 7], [0, 3, 3, -6], [1, 1, 4, -6], [2, 3, 4, -5]]]
[55, [[1, -1, 2, 7], [1, -2, 5, 5], [1, 3, 3, -6]]]
[56, [[0, 2, 4, -6]]]
[57, [[1, 2, 4, -6], [-3, 4, 4, 4]]]
[58, [[0, 0, -3, 7], [1, -2, -2, 7], [1, 4, 4, -5], [2, 2, 5, -5], [-2, 3, -3, 6]]]
[59, [[0, -1, 3, 7], [3, -3, 4, 5]]]
[60, [[1, -1, -3, 7], [1, 3, 5, -5]]]
[61, [[0, 0, -5, 6], [0, 3, 4, -6], [2, 2, -2, 7], [-2, 4, 4, -5]]]
[62, [[0, 1, 5, -6], [1, -3, -4, 6]]]
[63, [[1, 1, 5, -6], [1, -2, 3, 7], [2, -3, 5, 5], [3, 3, -3, 6]]]
[64, [[4, 4, 4, 4]]]
[65, [[0, 0, 1, 8], [0, 2, 5, -6], [2, -3, 4, 6]]]
[66, [[0, 1, -4, 7], [0, 4, 5, -5], [1, 2, -5, 6], [2, 2, 3, -7], [3, 4, 4, 5]]]
[67, [[0, -3, -3, 7], [1, 1, -1, 8], [-1, -1, 4, 7]]]
[68, [[1, 3, 3, -7], [3, 3, 5, 5]]]
[69, [[0, -1, 2, 8], [0, -2, 4, 7], [-1, 4, 4, -6], [2, -2, -5, 6]]]
[70, [[0, 3, -5, 6], [-1, -1, -2, 8], [1, 2, 4, -7], [2, 4, 5, 5], [3, 3, 4, 6]]]
[71, [[1, -3, 5, 6], [2, 3, 3, -7]]]
[72, [[0, 0, 6, -6], [0, -2, -2, 8], [2, 4, 4, 6]]]
[73, [[0, 1, 6, -6], [1, 2, -2, 8], [2, 2, 4, -7], [4, 4, -4, 5]]]
[74, [[0, -1, -3, 8], [0, 3, 4, -7], [1, -1, 6, -6], [2, 3, 5, 6]]]
[75, [[0, -1, -5, 7], [-1, -1, 3, 8], [1, 3, 4, -7], [3, -4, 5, 5]]]
[76, [[1, 1, 5, -7], [1, 5, 5, 5], [3, 3, 3, 7]]]
[77, [[0, -2, 3, 8], [-1, 2, 6, -6], [3, 4, -4, 6]]]
[78, [[0, 2, 5, -7], [1, -2, -3, 8], [1, 4, 5, 6], [2, 3, 4, 7]]]
[79, [[1, 2, 5, -7], [3, -3, -5, 6]]]
[80, [[0, 0, -4, 8], [2, 2, 6, 6]]]
[81, [[0, 0, 0, 9], [0, -3, 6, 6], [0, 4, 4, -7], [2, 2, -3, 8], [2, -4, 5, 6]]]
[82, [[1, -1, -4, 8], [1, 3, 6, 6], [1, 4, 4, 7], [2, 2, 5, 7], [4, -4, 5, -5]]]
[83, [[0, 1, -1, 9], [0, 3, 5, -7], [1, 3, -3, 8], [3, 3, -4, 7]]]
[84, [[1, 3, 5, 7]]]
[85, [[0, 0, -6, 7], [-1, -2, 4, 8], [-2, 3, 6, -6], [2, 4, -4, 7]]]
[86, [[0, 1, 6, -7], [0, 5, 5, 6], [2, 3, 3, 8], [3, -4, -5, 6]]]
[87, [[1, 1, -2, 9], [1, 1, 6, -7], [2, -3, -5, 7]]]
[88, [[0, 4, 6, 6], [2, 2, 4, 8]]]
[89, [[0, 2, -2, 9], [0, 2, 6, -7], [0, -3, 4, 8], [1, -4, 6, 6]]]
[90, [[0, 1, -5, 8], [0, 4, 5, 7], [-1, -2, -2, 9], [1, 2, 6, 7], [1, 3, 4, 8], [-2, 5, -5, 6], [3, -3, 6, -6], [3, 4, 4, -7]]]
[91, [[-1, -1, -5, 8], [1, -4, 5, 7], [3, 3, 3, -8], [4, 5, 5, -5]]]
[92, [[-1, -1, -3, 9], [3, 3, 5, -7]]]
[93, [[0, -2, -5, 8], [2, -2, -6, 7], [2, 3, 4, -8], [4, 4, -5, 6]]]
[94, [[0, -2, -3, 9], [0, 3, 6, 7], [1, 2, 5, 8], [2, 4, 5, -7]]]
[95, [[1, 2, -3, 9], [-1, 3, 6, -7], [3, 5, -5, 6]]]
[96, [[0, 4, 4, 8]]]
[97, [[1, 4, 4, -8], [2, 2, 5, -8], [-3, 4, 6, -6]]]
[98, [[0, 0, 7, -7], [0, -1, -4, 9], [0, 3, 5, 8], [-1, 5, 6, 6], [2, 2, 3, 9], [2, 3, 6, -7], [3, -3, -4, 8]]]
[99, [[0, 1, 7, -7], [0, 3, -3, 9], [1, 3, 5, -8], [3, 4, -5, 7]]]
[100, [[1, 1, 7, 7], [1, 3, 3, 9], [1, 5, 5, -7], [5, 5, -5, -5]]]
[101, [[0, 0, -1, 10], [0, -1, -6, 8], [0, -4, 6, 7], [2, -5, 6, 6]]]
[102, [[0, 2, 7, 7], [1, 1, 6, 8], [1, 2, 4, 9], [1, 4, 6, -7], [-2, 3, -5, 8], [4, -5, -5, 6]]]
[103, [[1, -1, -1, 10], [-1, 2, 7, -7], [-2, -3, -3, 9], [2, 5, -5, 7], [3, -3, -6, 7]]]
[104, [[0, 2, 6, 8], [4, -4, 6, -6]]]
[105, [[0, 1, -2, 10], [0, 4, 5, -8], [1, 2, 6, -8], [2, 2, -4, 9], [-2, 4, 6, -7]]]
[106, [[0, 0, -5, 9], [0, 3, 4, 9], [1, -4, -5, 8], [2, 2, 7, -7], [-3, -5, 6, 6], [-4, -4, 5, 7]]]
[107, [[1, 3, -4, 9], [3, 3, -5, 8]]]
[108, [[1, 1, 5, 9], [1, 3, 7, -7], [3, 3, 3, -9], [3, -5, -5, 7]]]
[109, [[0, 3, 6, -8], [-1, 2, -2, 10], [2, 4, -5, 8]]]
[110, [[0, 2, 5, 9], [0, 5, 6, -7], [1, -3, -6, 8], [2, 3, 4, -9], [3, -4, -6, 7]]]
[111, [[1, 1, -3, 10], [-1, -2, -5, 9], [1, -5, 6, 7], [-2, 3, 7, -7], [5, 5, 5, -6]]]
[112, [[2, 2, 2, 10], [-2, 6, 6, 6], [4, 4, 4, -8]]]
[113, [[0, 0, -7, 8], [0, 2, -3, 10], [0, 4, -4, 9], [2, -3, -6, 8], [4, 5, 6, -6]]]
[114, [[0, 1, 7, 8], [0, 4, 7, -7], [1, 2, 3, 10], [1, 4, 4, -9], [2, 2, 5, -9], [-2, 5, 6, 7], [3, 4, 5, -8]]]
[115, [[0, -3, -5, 9], [1, 1, 7, -8], [-1, -4, 7, 7], [1, 5, -5, 8], [3, 3, 4, -9], [4, -5, -5, 7]]]
[116, [[1, 3, 5, -9], [3, -3, 7, -7]]]
[117, [[0, 2, 7, -8], [-1, 4, 6, -8], [-2, -2, 3, 10], [2, 4, 4, -9], [3, 6, 6, -6], [4, 4, -6, 7]]]
[118, [[0, 1, 6, 9], [0, 3, 3, 10], [1, 1, 4, 10], [1, 2, -7, 8], [-2, 4, 7, 7], [2, 5, 5, -8], [3, 3, 6, -8]]]
[119, [[-1, -1, -6, 9], [-1, 3, -3, 10], [2, 3, 5, -9], [3, 5, -6, 7]]]
[120, [[0, 2, 4, 10], [2, 4, 6, -8]]]
[121, [[0, -2, -6, 9], [1, 2, -4, 10], [2, -2, -7, 8]]]
[122, [[0, 3, -7, 8], [0, 4, 5, -9], [1, 2, 6, -9], [-1, 6, 6, -7], [3, -4, -4, 9], [5, -5, 6, -6]]]
[123, [[0, -1, -1, 11], [0, -5, 7, 7], [-1, 3, 7, -8], [1, 4, 5, -9], [-3, 4, 7, -7], [3, -5, -5, 8]]]
[124, [[-1, 5, 7, -7], [3, -3, -5, 9]]]
[125, [[0, 0, -2, 11], [0, 3, -4, 10], [0, -5, 6, 8], [2, 2, 6, -9], [2, 6, -6, 7], [3, 4, -6, 8]]]
[126, [[0, 1, 5, 10], [0, 3, 6, -9], [1, -3, -4, 10], [1, 5, 6, -8], [2, 3, 7, -8], [-2, 4, 5, 9], [4, -5, -6, 7]]]
[127, [[1, -1, -2, 11], [1, 3, 6, -9], [-3, -3, -3, 10]]]
[128, [[0, 0, 8, 8]]]
[129, [[0, 1, 8, -8], [-2, -3, 4, 10], [2, 5, -6, 8]]]
[130, [[0, 0, 7, 9], [1, -1, 8, 8], [1, 2, 2, 11], [1, -2, -5, 10], [1, 4, 7, -8], [-2, 3, 6, 9], [-3, 6, 6, 7], [4, -4, 7, -7], [-4, 5, -5, 8]]]
[131, [[0, 1, -3, 11], [0, -1, -7, 9], [0, 5, 5, -9], [3, -3, -7, 8]]]
[132, [[1, 1, 3, 11], [1, 1, 7, -9], [1, -5, -5, 9], [-3, 5, 7, 7]]]
[133, [[0, 4, 6, -9], [-1, 2, 8, -8], [-1, 4, -4, 10], [2, -2, -2, 11], [2, 2, -5, 10], [-2, 4, 7, -8], [-5, 6, 6, -6]]]
[134, [[0, 2, 3, 11], [0, 2, 7, -9], [1, -4, -6, 9], [3, 3, 4, -10], [3, -5, -6, 8]]]
[135, [[-1, 2, -3, 11], [1, 2, 7, -9], [1, 3, -5, 10], [1, -6, 7, 7], [2, -5, -5, 9], [3, 3, -6, 9], [5, 5, 6, -7]]]
[136, [[0, 0, 6, 10], [0, 6, 6, -8], [2, 2, 8, -8], [2, 4, 4, -10]]]
[137, [[1, 6, -6, 8], [2, 4, -6, 9], [4, 6, 6, -7]]]
[138, [[0, 1, 4, 11], [0, 5, 7, -8], [1, -1, 6, 10], [1, 3, 8, -8], [-2, -2, -3, 11], [2, -2, 7, 9], [2, 3, 5, -10], [-2, 6, 7, -7], [3, -4, -7, 8], [4, 4, 5, -9]]]
[139, [[0, 3, 7, -9], [1, 1, -4, 11], [-1, -5, 7, 8], [4, 5, 7, -7]]]
[140, [[-1, 3, 3, 11], [1, -3, -7, 9], [3, 5, 5, -9]]]
[141, [[0, 2, -4, 11], [0, 4, -5, 10], [-1, -2, -6, 10], [-2, 3, 8, -8], [3, 4, 4, -10], [4, -5, -6, 8]]]
[142, [[-1, 2, 4, 11], [1, 4, 5, -10], [-2, 5, -7, 8], [3, 4, 6, -9]]]
[143, [[1, 5, -6, 9], [-2, 3, -3, 11], [2, -3, -7, 9], [3, 3, 5, -10], [3, 6, 7, -7]]]
[144, [[0, 4, 8, -8], [2, 2, 6, -10], [6, 6, -6, -6]]]
[145, [[0, 0, -8, 9], [0, -3, -6, 10], [2, 4, 5, -10], [-3, 6, 6, -8], [4, 4, -7, 8]]]
[146, [[0, 0, 5, 11], [0, 1, 8, -9], [0, -3, -4, 11], [1, 3, 6, -10], [2, 5, 6, -9], [3, -3, 8, 8], [5, -6, -6, 7]]]
[147, [[-1, -1, -1, 12], [1, 1, 8, -9], [-1, 3, -4, 11], [-1, 4, 7, -9], [3, 5, -7, 8]]]
[148, [[1, -1, 5, 11], [3, 3, 7, -9], [5, -5, 7, -7]]]
[149, [[0, -1, -2, 12], [0, 2, 8, -9], [0, -6, 7, 8], [2, 3, 6, -10]]]
[150, [[0, 1, -7, 10], [0, -2, -5, 11], [0, 5, 5, -10], [1, 1, 2, 12], [1, 2, -8, 9], [-1, 6, 7, -8], [-2, 3, 4, 11], [2, 4, 7, -9], [3, -4, -5, 10], [-4, 6, 7, 7], [-5, -5, 6, 8]]]
[151, [[-1, -1, -7, 10], [1, 2, -5, 11], [1, 5, 5, -10], [2, 7, 7, -7], [3, -5, -6, 9]]]
[152, [[0, 2, 2, 12], [0, 4, 6, -10], [4, -6, -6, 8]]]
[153, [[0, 0, -3, 12], [0, -2, -7, 10], [0, 6, -6, 9], [1, -2, -2, 12], [1, 4, 6, -10], [2, -2, -8, 9], [2, 6, -7, 8], [-3, 4, 8, -8]]]
[154, [[0, 1, 3, 12], [0, 3, -8, 9], [1, 2, 7, -10], [1, -4, -4, 11], [-1, 5, 8, -8], [1, 6, 6, -9], [2, -2, 5, 11], [3, -3, 6, 10], [4, -5, -7, 8]]]
[155, [[0, 3, -5, 11], [1, -1, -3, 12], [-1, 3, 8, -9], [-3, -3, 4, 11], [3, 4, -7, 9]]]
[156, [[1, -3, -5, 11], [1, 5, 7, -9], [-3, 7, 7, -7], [5, -5, -5, 9]]]
[157, [[2, 2, 7, -10], [-2, 4, -4, 11], [-2, -5, 8, 8], [-2, 6, 6, -9], [6, 6, 6, 7]]]
[158, [[0, -1, 6, 11], [0, 3, 7, -10], [-1, 2, 3, 12], [2, 3, 8, -9], [-3, 6, -7, 8], [-4, 5, 6, 9]]]
[159, [[1, 3, 7, -10], [-2, -3, -5, 11], [2, 5, -7, 9], [5, 6, 7, 7]]]
[160, [[0, 0, 4, 12], [4, -4, 8, 8]]]
[161, [[0, 1, -4, 12], [0, 5, 6, -10], [2, -2, -3, 12], [-3, -4, 6, 10], [5, 6, 6, 8]]]
[162, [[0, 0, 9, -9], [1, -1, 4, 12], [1, -2, 6, 11], [1, 4, 8, -9], [1, -5, -6, 10], [2, -3, 7, 10], [3, 4, 4, -11], [3, -6, -6, 9], [4, -4, 7, 9]]]
[163, [[0, 1, 9, -9], [-1, 4, -5, 11], [1, 7, -7, 8], [3, -3, -8, 9], [4, 7, 7, 7], [5, 5, 7, 8]]]
[164, [[1, -1, 9, -9], [3, 3, 5, -11], [3, -5, -7, 9]]]
[165, [[0, -1, -8, 10], [0, 4, 7, -10], [-1, 2, -4, 12], [-1, -6, 8, 8], [2, 2, -6, 11], [-2, 4, 8, -9], [2, -5, -6, 10], [4, 6, 7, 8]]]
[166, [[0, -2, 9, 9], [0, 6, 7, -9], [1, 1, 8, -10], [1, -4, -7, 10], [-2, 3, 3, 12], [2, 4, 5, -11], [-2, 7, 7, -8], [4, 5, 5, -10]]]
[167, [[-1, 2, 9, -9], [1, 3, -6, 11], [1, 6, -7, 9], [3, 3, -7, 10], [5, 5, 6, 9]]]
[168, [[0, 2, 8, -10], [2, -2, 4, 12], [-2, 6, 8, -8], [4, 4, 6, -10]]]
[169, [[1, 2, 8, -10], [2, 4, -7, 10], [4, 4, 4, -11], [4, 5, 8, 8], [4, 6, 6, 9]]]
[170, [[0, 0, -7, 11], [0, -1, 5, 12], [0, 5, 8, -9], [-1, -3, -4, 12], [2, 2, 9, -9], [2, 3, 6, -11], [3, -4, 8, 9], [3, 5, 6, -10], [6, -6, 7, -7]]]
[171, [[0, -5, -5, 11], [1, 1, -5, 12], [3, -3, -3, 12], [3, 4, 5, -11], [3, 7, 7, 8], [4, 5, 7, 9]]]
[172, [[1, 1, 1, 13], [1, -1, -7, 11], [1, 3, 9, -9], [1, 5, 5, -11], [-5, 7, 7, 7]]]
[173, [[0, 2, -5, 12], [0, 3, 8, -10], [0, 4, -6, 11], [-1, 6, 6, -10], [-2, 3, -4, 12], [3, 6, 8, 8]]]
[174, [[0, 1, 2, 13], [0, -2, 7, 11], [1, -2, 5, 12], [1, -3, 8, 10], [1, 4, 6, -11], [-2, 5, -8, 9], [3, 4, 7, -10], [5, -6, -7, 8]]]
[175, [[-1, -1, -2, 13], [-1, -2, -7, 11], [1, 5, -7, 10], [-2, 3, 9, -9], [2, 5, 5, -11], [3, 3, 6, -11], [3, 6, 7, 9], [5, 5, 5, 10]]]
[176, [[0, -4, -4, 12], [2, 6, 6, -10]]]
[177, [[0, -2, -2, 13], [0, -7, 8, 8], [2, -3, -8, 10], [2, 4, 6, -11], [4, 4, 8, 9], [4, 5, 6, 10]]]
[178, [[0, 0, 3, 13], [0, -3, -5, 12], [0, 4, 9, -9], [-1, 2, 2, 13], [2, 2, 7, -11], [2, 5, 7, -10], [3, -3, 4, 12], [4, -4, 5, 11], [-4, 7, -7, 8], [5, -5, 8, 8], [-5, 6, 6, 9]]]
[179, [[0, -1, -3, 13], [0, -3, -7, 11], [0, 7, -7, 9], [-1, 3, -5, 12], [3, 5, 8, 9]]]
[180, [[1, -1, 3, 13], [1, 3, 7, -11], [-1, 7, 7, -9], [3, -3, 9, -9], [3, -5, -5, 11], [5, -5, 7, 9]]]
[181, [[0, 0, -9, 10], [-1, 4, 8, -10], [2, 7, 8, 8], [3, 6, 6, 10], [4, 4, 7, 10]]]
[182, [[0, 1, 9, -10], [0, 5, 6, -11], [-1, -1, 6, 12], [-1, 6, 8, -9], [2, -3, 5, 12], [2, -4, 9, 9], [3, 3, 8, -10], [3, -4, 6, 11], [4, -6, -7, 9]]]
[183, [[1, 1, 9, -10], [1, -2, -3, 13], [1, 5, 6, -11], [2, 3, 7, -11], [2, 7, 7, 9], [3, 5, 7, 10], [-6, 7, 7, -7]]]
[184, [[0, -2, 6, 12], [2, 4, 8, -10]]]
[185, [[0, 0, -4, 13], [0, 2, 9, -10], [0, 6, -7, 10], [1, 2, -6, 12], [2, 6, 8, 9], [-3, 4, -4, 12], [6, -6, -7, 8]]]
[186, [[0, -1, 4, 13], [0, 1, -8, 11], [0, 4, 7, -11], [1, 2, -9, 10], [1, -4, -5, 12], [1, 6, 7, -10], [2, -2, 3, 13], [-3, 7, 8, -8], [4, -5, 8, 9], [5, -5, 6, 10]]]
[187, [[1, -1, -4, 13], [-1, -1, -8, 11], [1, 4, 7, -11], [-3, -3, -5, 12], [3, 4, 9, 9], [4, 5, 5, 11], [-5, 7, 7, -8]]]
[188, [[-1, 5, 9, -9], [3, -3, -7, 11]]]
[189, [[0, -2, -8, 11], [0, 3, -6, 12], [2, -2, -9, 10], [-2, 4, -5, 12], [2, 6, 7, 10], [3, 4, 8, 10], [4, 4, 6, 11], [-5, 6, 8, -8], [6, 6, 6, -9]]]
[190, [[0, 3, -9, 10], [1, -2, 4, 13], [1, 2, 8, -11], [1, -3, 6, 12], [1, 5, 8, -10], [2, -4, 7, 11], [-3, 6, -8, 9], [4, -5, 7, 10]]]
[191, [[-1, 3, 9, -10], [2, -3, -3, 13], [2, 5, 9, 9], [3, 5, 6, 11], [5, 6, 7, -9]]]
[192, [[4, 4, 4, -12]]]
[193, [[0, 6, 6, -11], [1, 8, 8, 8], [2, -2, -4, 13], [2, 2, 8, -11], [-2, -3, -6, 12], [2, 5, 8, 10]]]
[194, [[0, -1, -7, 12], [0, 3, 8, -11], [1, -6, -6, 11], [2, 3, 9, -10], [3, 4, 5, -12], [3, -6, -7, 10], [4, -4, 9, -9]]]
[195, [[0, 1, -5, 13], [0, 5, 7, -11], [1, 3, 8, -11], [-1, -5, -5, 12], [1, 7, 8, 9], [3, 4, 7, 11], [4, 7, 7, -9], [5, 5, 8, -9]]]
[196, [[-1, -1, 5, 13], [1, -5, -7, 11], [3, 3, -3, 13], [3, -5, 9, 9], [5, 5, 5, -11], [7, 7, -7, -7]]]
[197, [[-1, 4, -6, 12], [2, 6, 6, 11], [4, 6, 8, -9]]]
[198, [[0, 1, 1, 14], [0, -2, 5, 13], [0, 7, 7, -10], [1, -2, -7, 12], [1, 4, 9, -10], [2, -3, 4, 13], [-2, 3, -8, 11], [2, 5, 5, -12], [-2, 7, 8, -9], [3, 3, 6, -12], [3, -5, 8, 10], [4, 5, 6, -11], [6, -7, -7, 8]]]
[199, [[-1, 2, -5, 13], [1, 6, 9, 9], [1, 7, 7, 10], [2, 5, 7, 11], [3, 3, 9, 10]]]
[200, [[0, 0, 2, 14], [0, 0, 10, -10], [0, 6, 8, -10], [2, 4, 6, -12], [6, -6, 8, 8]]]