ガロア体GF(2^2), GF(2^3), GF(2^4)の加算表

手計算の時短のために加算表(addition table)を作りました。ご活用ください。
なお,表に出てくる  \alpha は原始多項式  f の根です。
もし間違いがありましたら,容赦なく指摘してくださると助かります。

GF(22)の加算表

原始多項式として  f(x)=x^2+x+1 を用いた, GF(2) の2次拡大体  GF(2^2) における加算表です。

f:id:hge:20180311005905p:plain

  •  \alpha^2 = \alpha + 1
  •  \alpha^3 = 1

GF(23)の加算表

原始多項式として  f(x)=x^3+x+1 を用いた, GF(2) の3次拡大体  GF(2^3) における加算表です。

f:id:hge:20180311005913p:plain

  •  \alpha^3= 1 + \alpha
  •  \alpha^4= \alpha + \alpha^2
  •  \alpha^5= 1 + \alpha + \alpha^2
  •  \alpha^6= 1 + \alpha^2
  •  \alpha^7= 1

GF(24)の加算表

原始多項式として  f(x)=x^4+x+1 を用いた, GF(2) の4次拡大体  GF(2^4) における加算表です。

f:id:hge:20180311005936p:plain

  •  \alpha^4= 1 + \alpha
  •  \alpha^5= \alpha + \alpha^2
  •  \alpha^6= \alpha^2 + \alpha^3
  •  \alpha^7= 1 + \alpha + \alpha^3
  •  \alpha^8= 1 + \alpha^2
  •  \alpha^9= \alpha + \alpha^3
  •  \alpha^{10}= 1 + \alpha + \alpha^2
  •  \alpha^{11}= \alpha + \alpha^2 + \alpha^3
  •  \alpha^{12}= 1 + \alpha + \alpha^2 + \alpha^3
  •  \alpha^{13}= 1 + \alpha^2 + \alpha^3
  •  \alpha^{14}= 1 + \alpha^3
  •  \alpha^{15}= 1


以上です。

正の整数を3つの正の整数の平方和・2つの正の整数の平方和へ分解する

前回の記事: 正の整数を4つの正の整数の平方和へ分解する - 冷水催眠

前回の記事の3つ及び2つの場合を記事にしました。


例えば, 41 という正の整数は  4^2 + 4^2 + 3^2,\ \ 6^2 + 2^2 + 1^2 というように3つの正の整数の平方和の形で2通りに表すことができます。
また, 41 = 5^2 + 4^2 というように2つの正の整数の平方和の形でも書くことができます。このように,与えられた正の整数  m に対し

 {
  m = a^2 + b^2 + c^2
}

もしくは

 {
  m = d^2 + e^2
}

を満たす正の整数の組  (a, b, c),\ (d, e) のリストをそれぞれ  1 \le m \le 250 の範囲で以下に列挙しました。以下のリストでは  \hspace{0.2 ex} a \ge b \ge c \hspace{0.2 ex},および  \hspace{0.2 ex} d \ge e \hspace{0.2 ex} を満たすように数字を並べました。ぜひぜひぜひぜひご活用ください。

リスト(3つの正の整数の平方和で表す)

[m] (a, b, c)
――――――――――
[3] (1, 1, 1)
[6] (2, 1, 1)
[9] (2, 2, 1)
[11] (3, 1, 1)
[12] (2, 2, 2)
[14] (3, 2, 1)
[17] (3, 2, 2)
[18] (4, 1, 1)
[19] (3, 3, 1)
[21] (4, 2, 1)
[22] (3, 3, 2)
[24] (4, 2, 2)
[26] (4, 3, 1)
[27] (3, 3, 3), (5, 1, 1)
[29] (4, 3, 2)
[30] (5, 2, 1)
[33] (4, 4, 1), (5, 2, 2)
[34] (4, 3, 3)
[35] (5, 3, 1)
[36] (4, 4, 2)
[38] (5, 3, 2), (6, 1, 1)
[41] (4, 4, 3), (6, 2, 1)
[42] (5, 4, 1)
[43] (5, 3, 3)
[44] (6, 2, 2)
[45] (5, 4, 2)
[46] (6, 3, 1)
[48] (4, 4, 4)
[49] (6, 3, 2)
[50] (5, 4, 3)
[51] (5, 5, 1), (7, 1, 1)
[53] (6, 4, 1)
[54] (5, 5, 2), (6, 3, 3), (7, 2, 1)
[56] (6, 4, 2)
[57] (5, 4, 4), (7, 2, 2)
[59] (5, 5, 3), (7, 3, 1)
[61] (6, 4, 3)
[62] (6, 5, 1), (7, 3, 2)
[65] (6, 5, 2)
[66] (5, 5, 4), (7, 4, 1), (8, 1, 1)
[67] (7, 3, 3)
[68] (6, 4, 4)
[69] (7, 4, 2), (8, 2, 1)
[70] (6, 5, 3)
[72] (8, 2, 2)
[73] (6, 6, 1)
[74] (7, 4, 3), (8, 3, 1)
[75] (5, 5, 5), (7, 5, 1)
[76] (6, 6, 2)
[77] (6, 5, 4), (8, 3, 2)
[78] (7, 5, 2)
[81] (6, 6, 3), (7, 4, 4), (8, 4, 1)
[82] (8, 3, 3)
[83] (7, 5, 3), (9, 1, 1)
[84] (8, 4, 2)
[86] (6, 5, 5), (7, 6, 1), (9, 2, 1)
[88] (6, 6, 4)
[89] (7, 6, 2), (8, 4, 3), (9, 2, 2)
[90] (7, 5, 4), (8, 5, 1)
[91] (9, 3, 1)
[93] (8, 5, 2)
[94] (7, 6, 3), (9, 3, 2)
[96] (8, 4, 4)
[97] (6, 6, 5)
[98] (8, 5, 3), (9, 4, 1)
[99] (7, 5, 5), (7, 7, 1), (9, 3, 3)
[101] (7, 6, 4), (8, 6, 1), (9, 4, 2)
[102] (7, 7, 2), (10, 1, 1)
[104] (8, 6, 2)
[105] (8, 5, 4), (10, 2, 1)
[106] (9, 4, 3)
[107] (7, 7, 3), (9, 5, 1)
[108] (6, 6, 6), (10, 2, 2)
[109] (8, 6, 3)
[110] (7, 6, 5), (9, 5, 2), (10, 3, 1)
[113] (9, 4, 4), (10, 3, 2)
[114] (7, 7, 4), (8, 5, 5), (8, 7, 1)
[115] (9, 5, 3)
[116] (8, 6, 4)
[117] (8, 7, 2), (10, 4, 1)
[118] (9, 6, 1), (10, 3, 3)
[120] (10, 4, 2)
[121] (7, 6, 6), (9, 6, 2)
[122] (8, 7, 3), (9, 5, 4)
[123] (7, 7, 5), (11, 1, 1)
[125] (8, 6, 5), (10, 4, 3)
[126] (9, 6, 3), (10, 5, 1), (11, 2, 1)
[129] (8, 7, 4), (8, 8, 1), (10, 5, 2), (11, 2, 2)
[131] (9, 5, 5), (9, 7, 1), (11, 3, 1)
[132] (8, 8, 2), (10, 4, 4)
[133] (9, 6, 4)
[134] (7, 7, 6), (9, 7, 2), (10, 5, 3), (11, 3, 2)
[136] (8, 6, 6)
[137] (8, 8, 3), (10, 6, 1)
[138] (8, 7, 5), (11, 4, 1)
[139] (9, 7, 3), (11, 3, 3)
[140] (10, 6, 2)
[141] (10, 5, 4), (11, 4, 2)
[142] (9, 6, 5)
[144] (8, 8, 4)
[145] (10, 6, 3)
[146] (9, 7, 4), (9, 8, 1), (11, 4, 3), (12, 1, 1)
[147] (7, 7, 7), (11, 5, 1)
[149] (8, 7, 6), (9, 8, 2), (12, 2, 1)
[150] (10, 5, 5), (10, 7, 1), (11, 5, 2)
[152] (10, 6, 4), (12, 2, 2)
[153] (8, 8, 5), (9, 6, 6), (10, 7, 2), (11, 4, 4)
[154] (9, 8, 3), (12, 3, 1)
[155] (9, 7, 5), (11, 5, 3)
[157] (12, 3, 2)
[158] (10, 7, 3), (11, 6, 1)
[161] (9, 8, 4), (10, 6, 5), (11, 6, 2), (12, 4, 1)
[162] (8, 7, 7), (11, 5, 4), (12, 3, 3)
[163] (9, 9, 1)
[164] (8, 8, 6), (12, 4, 2)
[165] (10, 7, 4), (10, 8, 1)
[166] (9, 7, 6), (9, 9, 2), (11, 6, 3)
[168] (10, 8, 2)
[169] (12, 4, 3)
[170] (9, 8, 5), (12, 5, 1)
[171] (9, 9, 3), (11, 5, 5), (11, 7, 1), (13, 1, 1)
[172] (10, 6, 6)
[173] (10, 8, 3), (11, 6, 4), (12, 5, 2)
[174] (10, 7, 5), (11, 7, 2), (13, 2, 1)
[176] (12, 4, 4)
[177] (8, 8, 7), (13, 2, 2)
[178] (9, 9, 4), (12, 5, 3)
[179] (9, 7, 7), (11, 7, 3), (13, 3, 1)
[180] (10, 8, 4)
[181] (9, 8, 6), (12, 6, 1)
[182] (10, 9, 1), (11, 6, 5), (13, 3, 2)
[184] (12, 6, 2)
[185] (10, 7, 6), (10, 9, 2), (12, 5, 4)
[186] (11, 7, 4), (11, 8, 1), (13, 4, 1)
[187] (9, 9, 5), (13, 3, 3)
[189] (10, 8, 5), (11, 8, 2), (12, 6, 3), (13, 4, 2)
[190] (10, 9, 3)
[192] (8, 8, 8)
[193] (11, 6, 6)
[194] (9, 8, 7), (11, 8, 3), (12, 5, 5), (12, 7, 1), (13, 4, 3)
[195] (11, 7, 5), (13, 5, 1)
[196] (12, 6, 4)
[197] (10, 9, 4), (12, 7, 2)
[198] (9, 9, 6), (10, 7, 7), (13, 5, 2), (14, 1, 1)
[200] (10, 8, 6)
[201] (10, 10, 1), (11, 8, 4), (13, 4, 4), (14, 2, 1)
[202] (12, 7, 3)
[203] (11, 9, 1), (13, 5, 3)
[204] (10, 10, 2), (14, 2, 2)
[205] (12, 6, 5)
[206] (10, 9, 5), (11, 7, 6), (11, 9, 2), (13, 6, 1), (14, 3, 1)
[209] (9, 8, 8), (10, 10, 3), (12, 7, 4), (12, 8, 1), (13, 6, 2), (14, 3, 2)
[210] (11, 8, 5), (13, 5, 4)
[211] (9, 9, 7), (11, 9, 3)
[212] (12, 8, 2)
[213] (10, 8, 7), (14, 4, 1)
[214] (13, 6, 3), (14, 3, 3)
[216] (10, 10, 4), (12, 6, 6), (14, 4, 2)
[217] (10, 9, 6), (12, 8, 3)
[218] (11, 9, 4), (12, 7, 5)
[219] (11, 7, 7), (13, 5, 5), (13, 7, 1)
[221] (11, 8, 6), (13, 6, 4), (14, 4, 3)
[222] (11, 10, 1), (13, 7, 2), (14, 5, 1)
[224] (12, 8, 4)
[225] (10, 10, 5), (11, 10, 2), (14, 5, 2)
[226] (9, 9, 8), (12, 9, 1)
[227] (11, 9, 5), (13, 7, 3), (15, 1, 1)
[228] (10, 8, 8), (14, 4, 4)
[229] (12, 7, 6), (12, 9, 2)
[230] (10, 9, 7), (11, 10, 3), (13, 6, 5), (14, 5, 3), (15, 2, 1)
[233] (12, 8, 5), (14, 6, 1), (15, 2, 2)
[234] (11, 8, 7), (12, 9, 3), (13, 7, 4), (13, 8, 1)
[235] (15, 3, 1)
[236] (10, 10, 6), (14, 6, 2)
[237] (11, 10, 4), (13, 8, 2), (14, 5, 4)
[238] (11, 9, 6), (15, 3, 2)
[241] (12, 9, 4), (13, 6, 6), (14, 6, 3)
[242] (12, 7, 7), (13, 8, 3), (15, 4, 1)
[243] (9, 9, 9), (11, 11, 1), (13, 7, 5), (15, 3, 3)
[244] (12, 8, 6)
[245] (10, 9, 8), (12, 10, 1), (15, 4, 2)
[246] (11, 10, 5), (11, 11, 2), (14, 5, 5), (14, 7, 1)
[248] (12, 10, 2), (14, 6, 4)
[249] (10, 10, 7), (11, 8, 8), (13, 8, 4), (14, 7, 2)
[250] (12, 9, 5), (15, 4, 3)

リスト(2つの正の整数の平方和で表す)

[m] (d, e)
――――――――――
[2] (1, 1)
[5] (2, 1)
[8] (2, 2)
[10] (3, 1)
[13] (3, 2)
[17] (4, 1)
[18] (3, 3)
[20] (4, 2)
[25] (4, 3)
[26] (5, 1)
[29] (5, 2)
[32] (4, 4)
[34] (5, 3)
[37] (6, 1)
[40] (6, 2)
[41] (5, 4)
[45] (6, 3)
[50] (5, 5), (7, 1)
[52] (6, 4)
[53] (7, 2)
[58] (7, 3)
[61] (6, 5)
[65] (7, 4), (8, 1)
[68] (8, 2)
[72] (6, 6)
[73] (8, 3)
[74] (7, 5)
[80] (8, 4)
[82] (9, 1)
[85] (7, 6), (9, 2)
[89] (8, 5)
[90] (9, 3)
[97] (9, 4)
[98] (7, 7)
[100] (8, 6)
[101] (10, 1)
[104] (10, 2)
[106] (9, 5)
[109] (10, 3)
[113] (8, 7)
[116] (10, 4)
[117] (9, 6)
[122] (11, 1)
[125] (10, 5), (11, 2)
[128] (8, 8)
[130] (9, 7), (11, 3)
[136] (10, 6)
[137] (11, 4)
[145] (9, 8), (12, 1)
[146] (11, 5)
[148] (12, 2)
[149] (10, 7)
[153] (12, 3)
[157] (11, 6)
[160] (12, 4)
[162] (9, 9)
[164] (10, 8)
[169] (12, 5)
[170] (11, 7), (13, 1)
[173] (13, 2)
[178] (13, 3)
[180] (12, 6)
[181] (10, 9)
[185] (11, 8), (13, 4)
[193] (12, 7)
[194] (13, 5)
[197] (14, 1)
[200] (10, 10), (14, 2)
[202] (11, 9)
[205] (13, 6), (14, 3)
[208] (12, 8)
[212] (14, 4)
[218] (13, 7)
[221] (11, 10), (14, 5)
[225] (12, 9)
[226] (15, 1)
[229] (15, 2)
[232] (14, 6)
[233] (13, 8)
[234] (15, 3)
[241] (15, 4)
[242] (11, 11)
[244] (12, 10)
[245] (14, 7)
[250] (13, 9), (15, 5)


以上です。
上の表から  \sqrt{6^2 + 3^2 + 2^2} = 7,\ \ \sqrt{7^2 + 4^2 + 4^2} = 9,\ \ \sqrt{7^2 + 6^2 + 6^2} = 11 といった式が作れますね。いろんなところで使えそうです!

他にも正の整数5つ,6つ,……の平方和や,0以上の数での平方和でもリストがあると便利だと思うのですが,こんなんばっかり記事にするわけにもいかないので悩ましい……