練習

一般的なもの

数の膨大さゆえ、特殊なものを除いて表記を$\arctan \frac{1}{x} = [x]$と簡略化する。また、断りがない限りすべて和は[1]=$\frac{\pi}{4}$とする。

公式は項数が少ない順、次に計算効率がいい順に記す。ただし、計算効率の評価法は一つではない。

著名な公式

発見年	公式名	公式
1706	マチンの公式	4[5]-[239]
1706	ハットンの公式	2[3]+[7]
1706	ヘルマンの公式	2[2]-[7]
1706	オイラーのマチン型公式	[2]+[3]
1863	ガウスのマチン型公式	12[18]+8[57]-5[239]
1844	シュトラスニツキーの公式	[2]+[5]+[8]
1764	オイラーのマチン型公式(2)	4[5]-[70]+[99]
1723	クリンゲンシェルナの公式	8[10]-[239]-4[515]
	シュテルマーの公式	6[8]+2[57]+[239]
1896	シュテルマーの公式(2)	44[57]+7[239]-12[682]+24[12943]
1982	高野喜久雄の公式	12[49]+32[57]-5[239]+12[110443]
1997	黄見利の公式	183[239]+32[1023]-68[5832]+ 12[110443]-12[4841182]-100[6826318]
2003	黄見利の公式(2)	183[239]+32[1023]-68[5832]+12[110321]- 100[6826318]-12[33366019650]+ 12[43599522992503626068]
2004	ウェザーフィールドの公式	83[107]+17[1710]-22[103697]-24[2513489]- 44[18280007883]+12[7939642926390344818]+ 22[3054211727257704725384731479018]

注:上から4つはマチンが発見、クリンゲンシェルナの公式の発見者はシムソン。

項数	公式	素因数
2	4[5] -[239]	13
	2[3] +[7]	5
	[2] +[3]	5
3	12[18] +8[57] -5[239]	5, 11
	8[10] -1[239] -4[515]	13, 101
	5[7] +4[68] +2[117]	5, 37
	5[7] +4[53] +2[4443]	5, 281
	5[7] +2[28] +2[443]	5, 157
	5[7] +2[27] +2[1068]	5, 73
	5[7] +2[26] -2[2057]	5, 677
	5[7] +2[23] -2[182]	5, 53
	3[7] +4[13] +2[38]	5, 17
	3[7] +4[11] -2[682]	5, 61
	4[6] +4[31] -1[239]	13, 37
4	44[57] +7[239] -12[682] +24[12943]	5, 13, 61
	20[57] +24[68] +12[117] -5[239]	5, 13, 37
	24[53] +20[57] -5[239] +12[4443]	5, 13, 281
	12[38] +20[57] +7[239] +24[268]	5, 13, 17
	24[29] -4[57] +7[239] -12[12238]	5, 13, 421
	22[28] +2[443] -5[1393] -10[11018]	5, 157, 197
	12[28] +20[57] -5[239] +12[443]	5, 13, 157
	12[27] +20[57] -5[239] +12[1068]	5, 13, 73
	12[27] +5[41] +10[46] +2[1068]	5, 29, 73
	17[27] +5[38] +10[557] +7[1068]	5, 17, 73
	12[26] +20[57] -5[239] -12[2057]	5, 13, 677
	17[26] +5[38] +3[2682] -7[8827]	5, 17, 677
	17[23] +8[182] +10[5118] +5[6072]	5, 53, 373
	12[23] +20[83] +8[182] -5[239]	5, 13, 53
	7[23] +10[30] +5[38] +3[182]	5, 17, 53
	17[22] +3[172] -2[682] -7[5357]	5, 61, 97
	10[22] +7[41] +12[75] +2[4193]	5, 29, 97
	16[21] +7[239] -4[616] +4[3141]	13, 17, 101
	16[21] +3[239] +4[421] +4[1985]	13, 17, 401
	16[21] +3[239] +4[251] -4[905]	13, 17, 109
	16[20] -1[239] -4[515] -8[4030]	13, 101, 401
	12[19] +8[57] -5[239] +12[343]	5, 13, 181
	10[19] +7[27] +2[1068] -5[3458]	5, 73, 181
	12[18] +8[109] +11[239] -8[6826318]	5, 13, 457
	12[18] +8[107] +8[122] -5[239]	5, 13, 229
	12[18] +8[99] +3[239] +8[307]	5, 13, 29
	12[18] +8[82] +8[187] -5[239]	5, 13, 269
	12[18] +8[73] +3[239] -8[2943]	5, 13, 41
	12[18] +8[67] -5[239] +8[382]	5, 13, 449
	12[18] +8[58] -5[239] +8[3307]	5, 13, 673
	12[18] +8[57] -10[577] -5[1393]	5, 13, 197
5	88[192] +39[239] +100[515] -32[1068] -56[173932]	5, 13, 73, 101
	88[172] +51[239] +32[682] +44[5357] +68[12943]	5, 13, 61, 97
	88[111] +7[239] -44[515] +32[682] +24[12943]	5, 13, 61, 101
	44[109] +95[239] -12[682] +24[12943] -44[6826318]	5, 13, 61, 457
	44[107] +44[122] +7[239] -12[682] +24[12943]	5, 13, 61, 229
	44[105] +32[117] +24[193] -5[239] -20[1893]	5, 13, 37, 149
	68[99] +27[239] -4[307] -12[12238] -24[58911]	5, 13, 29, 421
	56[99] +27[239] +32[307] +12[4193] -12[39307]	5, 13, 29, 97
	56[99] +39[239] +20[307] -24[2332] +12[6948]	5, 13, 29, 577
	44[99] +51[239] +44[307] -12[682] +24[12943]	5, 13, 29, 61
	56[99] +27[239] +20[307] +24[568] -12[19703]	5, 13, 29, 89
	56[99] +39[239] +8[307] +12[515] +24[3323]	5, 13, 29, 101
	44[99] +24[193] +27[239] +32[307] -12[20457]	5, 13, 29, 149
	56[99] +12[148] +27[239] +8[307] -12[34522]	5, 13, 29, 337
	68[99] -12[117] +39[239] +20[307] -24[882]	5, 13, 29, 37
6	322[577] +76[682] +139[1393] +156[12943] +132[32807] +44[1049433]	5, 13, 61, 89, 197
	122[319] +61[378] +115[557] +29[1068] +22[3458] +44[27493]	5, 17, 41, 73, 181
	100[319] +127[378] +71[557] -15[1068] +66[2943] +44[478707]	5, 13, 17, 41, 73
	337[307] -193[463] +151[4193] +305[4246] -122[39307] -83[390112]	5, 13, 17, 29, 97
	183[268] +32[682] +95[1568] +44[4662] -166[12943] -51[32807]	5, 13, 17, 61, 89
	183[268] +32[682] +95[1483] -7[9932] -122[12943] +51[29718]	5, 13, 17, 61, 761
	29[268] +269[463] +154[2059] +122[2943] -186[9193] +71[390112]	5, 13, 17, 41, 97
	241[268] -65[463] +32[682] -58[5357] -122[12943] -51[390112]	5, 13, 17, 61, 97
	100[268] +127[378] +27[1526] +71[2072] +51[2943] +115[16432]	5, 13, 17, 41, 257
	100[268] +83[378] +44[905] +115[1057] +51[2943] +71[3957]	5, 13, 17, 41, 109
	100[268] +83[378] +159[837] +7[2943] +71[247057] +44[740943]	5, 13, 17, 41, 317
	176[268] +7[343] +83[743] +32[8307] -115[12943] +44[27493]	5, 13, 17, 61, 181
	61[268] +122[343] +115[557] -32[1068] +83[3458] +44[27493]	5, 13, 17, 73, 181
	100[268] +115[327] +51[746] +12[1568] -71[4662] -32[157318]	5, 13, 17, 37, 89
	68[268] +115[307] +32[447] +95[1143] +51[34208] +39[44179]	5, 13, 17, 29, 53
	127[265] +100[515] +88[697] -32[1068] +39[2436] -56[173932]	5, 13, 37, 73, 101
	83[265] +100[268] +44[327] -12[557] -32[1068] +39[2436]	5, 13, 17, 37, 73
	227[255] -100[682] +44[2072] +51[2943] -27[12943] +88[16432]	5, 13, 41, 61, 257
	227[255] -56[682] -88[2059] +51[2943] +44[5357] -71[12943]	5, 13, 41, 61, 97
	183[255] +88[606] -56[682] +51[2943] -44[6118] -71[12943]	5, 13, 41, 53, 61
	183[255] +44[515] -56[682] +88[1303] +7[2943] -71[12943]	5, 13, 41, 61, 101
	127[241] +100[437] +44[2072] +24[2943] -12[16432] +27[28800]	5, 13, 41, 113, 257
	127[241] +100[437] -12[776] +24[1068] +44[2072] +39[28800]	5, 13, 73, 113, 257
	139[241] +88[437] -12[682] +44[2072] +24[12943] +51[28800]	5, 13, 61, 113, 257
	127[239] +68[1068] +232[1282] +44[2309] -88[4005] +56[4553]	5, 13, 53, 73, 389
	171[239] +68[993] -44[2633] +24[2943] +56[5593] -100[6826318]	5, 13, 37, 41, 457
	127[239] +200[818] +144[1303] -276[2943] -44[6118] -56[3014557]	5, 13, 41, 53, 101
	139[239] +120[682] +44[5357] +156[12943] +88[17923] +88[32807]	5, 13, 61, 89, 97
	139[239] +120[682] +132[5357] -88[5604] +244[12943] +176[2544482]	5, 13, 61, 97, 569
	139[239] +76[682] +88[1087] +112[12943] +88[32807] +44[534568]	5, 13, 61, 89, 149
	139[239] +32[682] +176[1087] -44[5357] +68[12943] -88[42658]	5, 13, 61, 97, 149
	183[239] -12[682] +44[1023] -44[5832] +24[12943] -88[6826318]	5, 13, 61, 229, 457
	95[239] +76[682] +176[882] +220[2917] -132[9466] +200[12943]	5, 13, 29, 37, 61
	95[239] +176[660] +76[682] +44[2621] +24[12943] -132[14942]	5, 13, 37, 61, 193
	127[239] +144[606] +36[5118] +56[6072] -8[6118] +76[52489]	5, 13, 41, 53, 373
	127[239] +200[606] -56[2072] -76[2943] -100[6118] -112[16432]	5, 13, 41, 53, 257
	139[239] +44[568] +76[682] +112[12943] +44[19703] +132[32807]	5, 13, 29, 61, 89
	83[239] +132[557] +68[1068] +144[1113] +100[9882] -44[21637]	5, 13, 17, 73, 733
	39[239] +176[557] +188[616] -32[1068] +100[3141] -144[173932]	5, 13, 17, 73, 101
	127[239] +100[515] +200[1303] -56[2072] -176[2943] -112[16432]	5, 13, 41, 101, 257
	127[239] +188[515] -120[1068] +88[41218] -144[173932] +88[3539232]	5, 13, 73, 101, 709
	127[239] +188[515] -120[1068] +88[37057] -144[173932] -88[409557]	5, 13, 73, 101, 149
	127[239] +188[515] -120[1068] +88[6105] -88[7181] -144[173932]	5, 13, 73, 101, 269
	127[239] +100[515] +56[1068] +88[1303] -176[2943] -56[173932]	5, 13, 41, 73, 101
	183[239] +44[515] -56[682] +88[6050] +24[12943] -88[6826318]	5, 13, 61, 101, 457
	183[239] -44[515] +32[682] +88[2057] +200[12943] -88[18107068]	5, 13, 61, 101, 677
	95[239] +44[515] +32[682] +176[782] +88[4030] +112[12943]	5, 13, 61, 101, 401
	95[239] +44[515] +32[682] +176[697] +24[12943] +88[74043]	5, 13, 37, 61, 101
	183[239] -44[500] +76[682] -88[3793] +200[12943] +132[32807]	5, 13, 53, 61, 89
	51[239] +220[477] +76[682] +44[5118] -88[6072] +68[12943]	5, 13, 53, 61, 373
	95[239] +132[437] -12[682] +88[1032] +44[3405] +68[12943]	5, 13, 29, 61, 113
	83[239] +144[437] +44[580] -12[776] +24[1068] +44[1710]	5, 13, 73, 113, 229
	95[239] +132[437] +44[580] -12[682] +44[1710] +24[12943]	5, 13, 61, 113, 229
	51[239] +132[421] +32[682] +176[782] -44[1985] +112[12943]	5, 13, 17, 61, 401
	71[239] +112[378] +44[616] +144[1303] -64[2943] +100[3141]	5, 13, 17, 41, 101
	71[239] +112[378] +144[606] -56[1772] -20[2943] -44[6118]	5, 13, 17, 41, 53
	71[239] +12[378] +144[489] +100[593] -20[2943] +44[165057]	5, 13, 17, 41, 541
	83[239] +100[353] +176[1113] -44[21637] -32[29193] -32[191693]	5, 13, 17, 617, 733
	83[239] +100[353] +144[1113] +64[2682] +32[8827] -44[21637]	5, 13, 17, 677, 733
	51[239] +132[353] +32[682] +176[1113] -64[12943] -44[21637]	5, 13, 17, 61, 733
	115[239] +100[353] -32[378] +144[1113] -64[2943] -44[21637]	5, 13, 17, 41, 733
	227[239] -44[348] -12[682] -44[1719] +68[12943] -88[6826318]	5, 13, 53, 61, 457
	27[239] +200[307] +68[5023] +56[12238] +68[32318] +44[58911]	5, 13, 29, 421, 461
	27[239] +200[307] +68[4193] +56[12238] -68[39307] +112[58911]	5, 13, 29, 97, 421
95[239] +132[307] -136[3307] -12[12238] +44[58911] -68[97643]	5, 13, 29, 421, 673
39[239] +188[307] +32[2332] -44[6948] +112[32318] -56[55368]	5, 13, 29, 461, 577
95[239] +132[307] -136[2332] +68[6948] +56[12238] +112[58911]	5, 13, 29, 421, 577
27[239] +200[307] -56[2072] +124[2943] +100[15179] -12[16432]	5, 13, 29, 41, 257
95[239] +132[307] -68[1537] +56[8277] -124[29307] -12[34522]	5, 13, 29, 337, 701
95[239] +132[307] -68[1292] +56[12238] +68[12943] +44[58911]	5, 13, 29, 61, 421
83[239] +88[307] +68[1068] +100[1282] +56[4553] -44[17620]	5, 13, 29, 73, 389
95[239] +76[307] +100[882] +68[2621] -56[14942] +124[25943]	5, 13, 29, 37, 193
95[239] +132[307] -34[682] +90[12943] +22[34522] +22[106007]	5, 13, 29, 61, 337
139[239] +88[307] -56[682] -44[5357] +68[12943] +88[39307]	5, 13, 29, 61, 97
95[239] +132[307] -12[682] -44[2482] -44[3671] +68[12943]	5, 13, 29, 61, 293
95[239] +132[307] -56[682] +88[2332] -44[6948] +112[12943]	5, 13, 29, 61, 577
95[239] +132[307] -12[682] -44[1902] -44[6687] +68[12943]	5, 13, 29, 61, 409
95[239] +132[307] -56[682] +44[1607] +44[5928] +68[12943]	5, 13, 29, 61, 137
95[239] +176[307] -56[682] -88[800] +68[12943] +44[29718]	5, 13, 29, 61, 761
27[239] +132[307] +136[568] -12[19703] +56[27823] +56[34522]	5, 13, 29, 89, 337
27[239] +132[307] +136[568] +56[12238] -68[19703] +112[58911]	5, 13, 29, 89, 421
83[239] +132[307] -32[568] +112[1568] -56[6107] +44[19703]	5, 13, 17, 29, 89
27[239] +132[307] +80[568] +112[1123] +44[19703] -56[160590]	5, 13, 29, 89, 109
83[239] +76[307] +80[568] +56[1113] -12[19703] +56[4180652]	5, 13, 29, 89, 733
95[239] +64[307] +68[515] +136[3323] +56[12238] +112[58911]	5, 13, 29, 101, 421
39[239] +176[307] +12[515] +56[1068] -88[3323] -56[173932]	5, 13, 29, 73, 101
95[239] +176[307] -44[515] -56[682] -88[3323] +112[12943]	5, 13, 29, 61, 101
39[239] +144[307] +32[447] +68[1068] +44[2309] -56[48737]	5, 13, 29, 53, 73
27[239] +132[307] +68[443] +68[813] +56[12238] +44[58911]	5, 13, 29, 157, 421
83[239] +88[307] +56[418] +68[4193] +44[39307] +56[333367]	5, 13, 29, 97, 241
27[239] +132[307] +68[392] +68[1068] +56[12238] +44[58911]	5, 13, 29, 73, 421
139[239] +44[301] +32[682] +112[12943] +88[32807] -44[70543]	5, 13, 61, 89, 509
95[239] +88[294] +32[682] +88[2322] +68[12943] -44[29718]	5, 13, 61, 109, 761
51[239] +132[294] +32[682] +44[905] +88[3957] +68[12943]	5, 13, 17, 61, 109
95[239] +88[294] +44[515] +32[682] -88[1929] +24[12943]	5, 13, 61, 101, 109
71[239] +112[293] +44[616] +132[3141] -32[12238] -64[13443]	5, 13, 17, 101, 421
39[239] +144[293] +44[616] +64[2682] +100[3141] +32[8827]	5, 13, 17, 101, 677
71[239] +112[293] +12[616] +64[1432] +132[3141] -32[265842]	5, 13, 17, 101, 193
39[239] +144[293] +12[616] +64[807] +100[3141] +32[246147]	5, 13, 17, 101, 521
7[239] +176[293] +44[616] +32[682] +132[3141] -64[12943]	5, 13, 17, 61, 101
39[239] +112[293] +64[307] +12[616] +68[3141] -32[3323]	5, 13, 17, 29, 101
27[239] +136[273] +64[307] -68[2072] -12[12238] -24[58911]	5, 13, 29, 257, 421
27[239] +124[273] +76[307] -56[2072] -12[5182] +12[2794602]	5, 13, 29, 257, 661
27[239] +112[273] +88[307] -56[2072] +12[4193] -12[39307]	5, 13, 29, 97, 257
39[239] +100[273] +88[307] -56[2072] -12[3307] -12[177057]	5, 13, 29, 257, 673
39[239] +112[273] +76[307] -56[2072] -24[2332] +12[6948]	5, 13, 29, 257, 577
51[239] +88[273] +88[307] -12[682] -44[2072] +24[12943]	5, 13, 29, 61, 257
27[239] +112[273] +76[307] +24[568] -56[2072] -12[19703]	5, 13, 29, 89, 257
39[239] +112[273] +64[307] +12[515] -56[2072] +24[3323]	5, 13, 29, 101, 257
51[239] +88[269] +88[477] +32[682] +44[5357] +68[12943]	5, 13, 61, 97, 373
115[239] +68[268] +108[1958] +12[2928] -64[7193] +32[228382]	5, 13, 17, 349, 593
83[239] +100[268] +44[1958] +64[2682] +44[2928] +32[8827]	5, 13, 17, 349, 677
51[239] +164[268] -52[1568] +12[4662] -64[6898] -32[534568]	5, 13, 17, 89, 149
83[239] +100[268] +44[1568] +64[2682] +44[4662] +32[8827]	5, 13, 17, 89, 677
83[239] +132[268] -64[1458] -20[1568] -32[4443] +44[4662]	5, 13, 17, 89, 281
83[239] +100[268] +64[1432] +44[1958] +12[2928] -32[5099]	5, 13, 17, 193, 349
51[239] +132[268] +108[1292] -32[12238] -20[12943] +44[1000267]	5, 13, 17, 61, 421
83[239] +100[268] +76[1282] +32[8832] +44[26337] +12[29193]	5, 13, 17, 389, 617
115[239] +68[268] +44[905] +20[3957] -32[12238] -64[114483]	5, 13, 17, 109, 421
115[239] +68[268] +44[905] -12[3957] +32[6507] -32[6826318]	5, 13, 17, 109, 457
83[239] +100[268] +44[905] +64[2682] -44[3957] +32[8827]	5, 13, 17, 109, 677
83[239] +100[268] +32[807] +12[1568] +32[3882] +44[4662]	5, 13, 17, 89, 521
83[239] +100[268] +32[767] +12[1568] +44[4662] +32[5182]	5, 13, 17, 89, 661
51[239] +132[268] +32[682] +44[2307] +44[2928] -20[12943]	5, 13, 17, 61, 349
51[239] +132[268] +32[682] +44[1432] -20[12943] +44[13043]	5, 13, 17, 61, 193
51[239] +132[268] +76[682] -44[1407] -20[12943] +44[51412]	5, 13, 17, 61, 137
51[239] +132[268] +32[682] +44[1282] -20[12943] -44[198505]	5, 13, 17, 61, 389
51[239] +132[268] +32[682] +44[1087] -44[6898] -20[12943]	5, 13, 17, 61, 149
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83[239] +144[268] -12[557] -76[1068] -44[6898] -44[409557]	5, 13, 17, 73, 149
83[239] +100[268] +32[557] -32[1068] +44[1958] +44[2928]	5, 13, 17, 73, 349
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83[239] +144[268] -56[557] -32[1068] +44[1432] +44[75382]	5, 13, 17, 73, 193
83[239] +144[268] -12[557] -32[1068] -44[1118] -44[5283]	5, 13, 17, 73, 173
83[239] +100[268] +32[557] +44[905] -32[1068] -44[3957]	5, 13, 17, 73, 109
83[239] +144[268] -12[557] -44[863] -32[1068] +44[13332]	5, 13, 17, 73, 337
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61[239] +122[268] +22[463] +32[557] -32[1068] -22[390112]	5, 13, 17, 73, 97
83[239] +132[268] -32[443] +12[1568] +44[4662] +32[43932]	5, 13, 17, 89, 157
83[239] +112[268] +20[421] -44[1985] -32[4443] +64[30857]	5, 13, 17, 281, 401
83[239] +144[268] -44[421] -44[1985] +64[2682] +32[8827]	5, 13, 17, 401, 677
83[239] +144[268] -44[421] +32[557] -32[1068] -44[1985]	5, 13, 17, 73, 401
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39[239] +100[268] +88[395] +32[557] -32[1068] -44[37107]	5, 13, 17, 73, 353
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115[239] +100[268] -32[378] +44[1958] +44[2928] -64[2943]	5, 13, 17, 41, 349
71[239] +100[268] +12[378] +44[565] -20[2943] +44[3557]	5, 13, 17, 41, 229
71[239] +100[268] +12[378] +44[483] -20[2943] -44[51693]	5, 13, 17, 41, 569
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83[239] +144[268] -44[343] +64[2682] +32[8827] +44[27493]	5, 13, 17, 181, 677
71[239] +100[268] +24[327] +52[1432] +20[2621] -32[14942]	5, 13, 17, 37, 193
71[239] +100[268] +24[327] +20[1252] +32[2855] +52[3557]	5, 13, 17, 37, 229
51[239] +132[268] +44[327] -44[438] +32[682] -20[12943]	5, 13, 17, 37, 61
71[239] +100[268] +44[327] +12[378] -44[993] -20[2943]	5, 13, 17, 37, 41
51[239] +100[268] +64[307] -20[1958] +12[2928] -32[9210]	5, 13, 17, 29, 349
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39[239] +68[268] +64[307] +56[395] +32[1723] -12[37107]	5, 13, 17, 29, 353
71[239] +100[268] +44[307] +12[378] -44[829] -20[2943]	5, 13, 17, 29, 41
51[239] +80[268] +64[307] +20[343] +32[5087] +12[27493]	5, 13, 17, 29, 181
39[239] +68[268] +64[307] +56[327] -12[882] +44[18543]	5, 13, 17, 29, 37
51[239] +64[242] +68[268] +44[905] +20[3957] +32[219602]	5, 13, 17, 53, 109
51[239] +44[242] +88[268] +32[682] +44[2673] -20[12943]	5, 13, 17, 53, 61
51[239] +64[242] +80[268] +8[659] -20[6443] +12[1610057]	5, 13, 17, 53, 241
51[239] +76[242] +68[268] +8[642] -32[5118] -12[6072]	5, 13, 17, 53, 373
39[239] +88[242] +56[268] +32[557] +12[1068] -44[2309]	5, 13, 17, 53, 73
51[239] +64[242] +68[268] +32[500] -20[1568] +12[4662]	5, 13, 17, 53, 89
7	1587[2852] +295[4193] +593[4246] +359[39307] +481[55603] +625[211050] -708[390112]	5, 13, 17, 29, 97, 433
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	481[795] +295[4193] +256[4246] +625[13307] +144[14256] -266[39307] -227[390112]	5, 13, 17, 29, 97, 641
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	1337518[4417548] +4733818[8082212] -890639[8188568] +1029535[12477035] -593450[18975991] +566594[22709274] -1270949[24208144] +39581[25010458] +1126917[53798468] -1746166[63199427] -1774677[201229582] -1126917[231373438] +407298[462333568] -216308[2189376182]	5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 181, 269, 281
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	1337518[4417548] +3606901[8082212] -2017556[8188568] +2253834[8538694] +2156452[12477035] -593450[18975991] +566594[22709274] -2397866[24208144] +39581[25010458] -1746166[63199427] -1126917[103224943] -2901594[201229582] +1534215[462333568] -1343225[2189376182]	5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 181, 241, 281
	1337518[4417548] +1126917[6225244] +349353[8188568] +1239992[11918668] +7003345[12477035] +533467[18975991] -3040307[22709274] -4877850[24208144] -2327328[25010458] +620743[63199427] -4141586[201229582] +1534215[462333568] +3390593[2189376182] -3606901[18986886768]	5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 181, 281
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	2464435[4417548] -1126917[4640657] +5860735[8082212] -2017556[8188568] -97382[12477035] -593450[18975991] +1693511[22709274] -1270949[24208144] +1166498[25010458] -1746166[63199427] -1774677[201229582] -719619[462333568] -216308[2189376182] +2253834[29546599818]	5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 181, 281, 409
	2464435[4417548] -2775701[4640657] +1746166[4956193] +4017187[6456617] -2017556[8188568] +1055334[18975991] +973892[22709274] +377835[24208144] -579668[25010458] +622237[92985095] -125893[201229582] +1529858[2189376182] -1648784[7830729512] +1534215[29546599818]	5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 281, 409, 733
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	940633[4218932] -801522[4641970] -1241486[5982670] -2040418[6315768] -2099818[10845307] -977739[32944452] -2938956[34713882] +321201[60033932] -1024216[160007778] -384106[244653118] +2263756[361632045] -1987150[2189376182] -500025[2701984943] +1673970[5475957057]=-[1]	5, 13, 17, 29, 41, 53, 61, 73, 89, 101, 149, 157, 233
	1534215[4218932] -1438183[4417548] +1241486[5982670] +2368403[6225244] -703015[9548768] +619249[11918668] +431625[15165443] +2239705[22709274] -29463[24208144] +533467[24660332] -172370[25010458] +3037778[82323393] +789073[201229582] -919323[2189376182]	5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 157, 281
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	1534215[4180652] +1241486[4238657] +930220[4417548] -1029535[4956193] +758145[8188568] -186152[18975991] -560323[22709274] +377835[24208144] +954547[25010458] +914966[92985095] -125893[201229582] +1241486[757346975] -4357[2189376182] -407298[7830729512]	5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 281, 421, 733
	593633[4172561] -1114192[4218932] -2043008[5982670] +188986[6315768] -955057[9548768] -1637409[10496132] -2021515[16862243] +935878[60033932] +384106[63199427] -293620[160007778] +973892[361632045] +33310[376597307] +2368005[1735194643] -1185628[14033378718]=-[1]	5, 13, 17, 29, 41, 53, 61, 73, 101, 149, 157, 181, 709
	977739[4172561] -2077524[4218932] -801522[4641970] -1241486[5982670] -321201[9548768] +261801[10496132] -3018157[16862243] +1424618[34713882] -8014[64180002] -384106[244653118] +754401[1735194643] +1292808[2189376182] +837378[2701984943] -589786[5475957057]=-[1]	5, 13, 17, 29, 41, 53, 61, 73, 89, 101, 157, 233, 709
	3606901[4055901] -2269383[4417548] -1534215[5926518] +1126917[6225244] +1262825[8188568] +619249[11918668] +410286[12100093] +4140368[18975991] +566594[22709274] +1504752[24208144] -1706585[25010458] +1620273[201229582] +1241486[482786676] -216308[2189376182]	5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 281, 317
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15	5034126[20942043] +1546003[22709274] +6478634[24185182] +688806[24208144] -980346[27005643] +1573796[54610269] +4414877[63199427] +1553394[100656977] +6658985[115453918] +4676954[116428593] -795268[201229582] +5996230[909140573] +4818755[2189376182] +1980157[18986886768] +1337518[250645741818]	5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 181, 337, 389
	5345097[18975991] +6293190[22709274] +4416066[24208144] +3338499[33663593] -1948675[36239416] -2504016[56824828] +1802617[168623905] +4238858[201229582] +838628[225823278] -2911314[227661182] +6397856[545216547] +2479984[657922943] +3370623[934981432] +926158[2189376182] +4944419[7804016832]	5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 109, 233, 277, 557
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	5752395[18975991] +1808080[22709274] +9913165[24185182] -648712[24208144] -5938842[52630807] +2193045[54610269] +6869062[75250343] +5200686[115453918] -7587331[168623905] +2282091[201229582] -4772049[284862638] -619249[546365045] +6091325[909140573] +1665766[2189376182] -2815282[164432798314]	5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 109, 389, 541
	6371644[18975991] +1188831[22709274] +6478634[24185182] -648712[24208144] +5391298[50693817] -1337518[51853693] +1573796[54610269] -619249[63199427] -407298[100656977] +6750155[115453918] -1152440[201229582] +6710574[909140573] -838545[2189376182] +4053780[5127511733] +2603331[18986886768]	5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 181, 389, 461
	6371644[18975991] +1808080[22709274] +12109198[24185182] -4083243[24208144] +619249[42536057] +7587331[50693817] -3533551[51853693] -10091642[52630807] +2193045[54610269] -3434531[57009056] +17799332[115453918] -3348473[201229582] +11721889[909140573] +8633848[2189376182] -2196033[3508373380]	5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 197, 389, 461
	5752395[18975991] +1808080[22709274] +9293916[24185182] -3463994[24208144] -5011315[38105751] +7587331[50693817] +2097013[51853693] -3841829[52630807] +4389078[54610269] +11549519[115453918] -3348473[201229582] +8287358[909140573] +619249[1075411628] +5199317[2189376182] +2815282[223632092120]	5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 149, 389, 461
	8567677[18975991] +2427329[22709274] +5859385[24185182] -2844745[24208144] +3434531[27005643] -2504311[52630807] +1573796[54610269] +3342939[115453918] -3533551[168623905] -533191[201229582] -4772049[284862638] +619249[367829623] +4233578[909140573] -2388014[2189376182] -2815282[250645741818]	5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 109, 337, 389
	1870985[18975991] +3477908[22709274] +6420833[22997655] -2860294[24208144] +9469110[25010458] -1629952[27089889] -4248832[30808747] +6518215[34713882] +523217[63199427] -1126917[193788912] +2027276[201229582] -3394655[462333568] -3127622[2189376182] +1434900[4006581229] +6965094[37093513413]	5, 13, 17, 29, 37, 53, 61, 73, 89, 97, 101, 181, 233, 281
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	5874226[18975991] +8305600[21638297] +4235409[22709274] +8226445[24185182] +758363[24208144] +6249813[24786723] +7488311[41151258] -12496933[52630807] +3766841[54610269] +4432051[115453918] +1608654[201229582] +619249[257926007] -36484[909140573] +8962096[2189376182] +3294285[3508373380]=2[1]	5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 197, 389, 653
	1860735[17546316] +3771301[18975991] +2368403[21638297] +2254959[22709274] +3078548[24208144] +1746166[24789357] +503330[27872057] +1981094[47927722] +3027233[107258455] -2931201[127120947] +1543935[201229582] +426890[203420807] +171020[303690197] -1553826[2189376182] +2020675[30687656693]	5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 113, 197, 229, 449
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26	152014292229[392943720343] +60325885083[476704288228] -164945727087[529611885793] +147824211317[548275361132] +65851303444[557574720396] -36019205899[658659070243] +269786272473[751830514193] +94173054674[764180902035] -546822892588[819027087393] +374275735145[1029114662298] +147067592325[1214163909557] -69430327926[1512358008568] +82589794225[2517728913757] -95115760914[3160683755650] +91055060163[8600785841874] -240303291743[10463035477341] +99429583338[10818367923318] +92477026812[16326085782557] +53025781549[20984513530807] -202628175590[22486542112682] -183568823263[55707370528529] +108074146783[69971515635443] -51240615943[109805699900318] +79982845860[121718582464579] +1296083170[155044900887297] -103240793335[1367017926441055]	5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 173, 181, 193, 197, 229, 257, 269, 277, 577, 653
	246005956384[360562783237] -112112075621[392943720343] +69903671507[427792074403] +61519280807[476704288228] +10577237849[503738737569] -3957524828[529611885793] +60324224077[574689103867] -15243250005[578998796618] +3939164944[588318842848] -118933132748[764180902035] +73449763129[804006285737] +54871315233[824614976145] -48049843200[908173846068] -4516713229[2203180357932] +69828732869[2292939666693] -19750313928[4426895335369] +69819345104[6696078931715] +65730284497[6988977618513] +105998623218[10818367923318] -22320488903[13253577991727] -77009733944[21881002771243] +5884751286[26573652775818] -106476683528[139889483331112] -11733408764[155044900887297] -1569212338[6604584685078343] +48204000632[17251727577487177]	5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 157, 181, 197, 229, 233, 241, 269, 313, 353, 577
	190512971108[324363564309] -182115180498[461903069803] -183973243237[476704288228] +162063041690[553287903212] +246770542275[564006794307] -473637972094[588318842848] -581410484560[596264409557] -631237614964[597964988589] -194958715463[764180902035] +235582892126[951210399823] +203499594693[985181320882] -39025407023[1017179464093] -409141803662[1991134822932] -391762115421[2517728913757] +559763918580[3160683755650] -276026640176[3215723457318] +365549375353[8762990799852] +120781821440[10818367923318] +293873481852[14483848717682] -155608779061[16248096944867] -754564811846[61849783303008] -21075274548[69971515635443] +71789820562[169838669284032] -300969442548[1367017926441055] -447442721792[2799978903689557] -115480230498[3386696477716649]=-3[1]	5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 173, 181, 193, 229, 241, 277, 293, 577, 601
	242772722145[324363564309] -239339538150[369618065903] -147811812370[422771493022] -237347353474[461903069803] +154580008240[476704288228] -129772867159[548275361132] -186058229822[557574720396] +13203257628[564006794307] -323478422662[764180902035] -172791804075[833843894645] +99265829615[951210399823] -30389245032[1512358008568] -187195858330[1991134822932] -81515799704[2517728913757] +179343800140[3160683755650] +186364486757[10818367923318] -30045708839[11896361530243] +143188474315[14483848717682] +33870019893[16248096944867] +102179411086[16326085782557] -190753045191[69971515635443] +133295074507[600662758381590] +89749325559[686308367425978] -156815876465[1367017926441055] -203255891509[2799978903689557] -21670383406[6805164953551432]=-2[1]	5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 173, 181, 193, 197, 229, 257, 277, 293, 577
	38763569175[324241877809] +755595651182[324363564309] +64718295512[461903069803] +1193351502707[476704288228] +104162339322[516500901401] +1208869781622[548275361132] -52813030512[564006794307] -391997542676[615486765808] -456916335525[764180902035] -91084715907[833843894645] +614796055868[951210399823] -129679970505[1005787402861] -432128722276[1991134822932] -726798729016[2465414983868] -544613420160[2517728913757] +225719319450[2766528885637] +1237805216694[3160683755650] -640117195392[7167325038957] +66199739033[10818367923318] +208546053484[14483848717682] +6654657243[16248096944867] +461958974632[16326085782557] -102355555343[69971515635443] +927451337454[686308367425978] -441871648238[1367017926441055] -247705754204[2799978903689557]=8[1]	5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 173, 181, 193, 229, 257, 277, 293, 557, 577
27	166553022292[970522492753] +222363417479[989193552378] -134276698825[1096452832428] +168215423310[1280283860113] +75023059326[1341087111018] +136852193784[1689015353762] +217055842606[1822081215762] +103141369176[2184607268277] +28713480349[2278678014557] +221440571852[2635662131192] +184010343804[3165256360443] -130014434756[3385630462882] +30039704433[4426171412662] -125016355012[4963640229982] -268445832064[4972090102688] +80047317279[6306451059345] +229618316915[10221155603807] +30192504858[10305371319950] +18293883503[13688849577057] -44291036474[14483848717682] +29376832104[24632166555862] -139440534748[39537374317540] -59815251609[69971515635443] +62403552219[104225908824307] +59060238669[106851921608307] -169497968425[169838669284032] -238261971358[452493528674723]	5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 109, 113, 137, 149, 173, 181, 229, 241, 257, 269, 277, 281, 317, 541, 601

Ma=ジョン・マチン

Si=ロバート・シムソン

Sto=フレドリック・カール・シュテルマー

Eu=レオンハルト・オイラー

Ga=カール・フリードリヒ・ガウス

Wr=ジョン・ウィリアム・レンチ

Str=レオポルド・カール・シュルツ・フォン・シュトラスニツキー

Sc=カール・ハインリヒ・シェルバッハ

Lo=シドニー・ラックストン・ローニー

Es=エドワード・ブラインド・エスコット

Le=デリック・ヘンリー・レーマー

Be=アルバート・アーノルド・ベネット

Sh=カール・ハインリヒ・シェルバッハ

Ta=高野喜久雄

Sh=柴田昭彦

Mu=村田健郎

Hw=黄見利

特殊なもの

$\arcsin x = \sum_{n=0}^\infty \frac{(2n-1)!!}{(2n)!!} \frac{x^{2n+1}}{2n+1}$を利用

$\frac{\pi}{3} = 2 \arcsin \frac{1}{2} = 2\sum_{n=0}^\infty \frac{(2n-1)!!}{(2n)!!} \frac{1}{2^{2n+1}(2n+1)}$(ニュートンの公式)

$= \sum_{n=0}^\infty \frac{(2n-1)!!}{(2n)!!} \frac{1}{2^{2n}(2n+1)} = \sum_{n=0}^\infty \frac{((2n-1)!!)^2}{(2n)!} \frac{1}{2^{2n}(2n+1)}$

$= \sum_{n=0}^\infty \frac{((2n-1)!!)^2}{(2n+1)!} \frac{1}{2^{2n}} = \sum_{n=0}^\infty \frac{((2n-1)!!)^2}{(4n+2)!!}$(松永良弼の公式)

$\frac{\arcsin x}{\sqrt{1-x^2}} = \sum_{n=0}^\infty \frac{2^{n} n!}{(2n-1)!!} \frac{x^{2n+1}}{2n+1}$を利用

$\frac{\pi}{2} = \sqrt{2}\frac{\arcsin \frac{1}{\sqrt{2}}}{\sqrt{1-(\frac{1}{\sqrt{2}})^2}} = \sqrt{2}\sum_{n=0}^\infty \frac{2^{n} n!}{(2n-1)!!} \frac{(\frac{1}{\sqrt{2}})^{2n+1}}{2n+1} = \sum_{n=0}^\infty \frac{n!}{(2n+1)!!}$

$\frac{2\sqrt{3}\pi}{9} = 2\frac{\arcsin \frac{1}{2}}{\sqrt{1-(\frac{1}{2})^2}} = 2\sum_{n=0}^\infty \frac{2^{n} n!}{(2n-1)!!} \frac{(\frac{1}{2})^{2n+1}}{2n+1} = \sum_{n=0}^\infty \frac{n!}{2^n(2n+1)!!} = \sum_{n=0}^\infty \frac{(n!)^2}{(2n+1)!}$

$\frac{\pi}{5\sqrt{\phi +2}} = \frac{\arcsin \sin{\frac{\pi}{10}}}{\sqrt{1-\sin^2{\frac{\pi}{10}}}} = 2\sum_{n=0}^\infty \frac{(n!)^2}{\phi^{2n+1}(2n+1)!}$

$\arcsin^2 x = 2\sum_{n=0}^\infty \frac{2^{2n} (n!)^2}{(2n+2)!}x^{2n+2}$を利用

$\frac{\pi^2}{9} = 4\arcsin^2 \frac{1}{2} = \sum_{n=0}^\infty \frac{(n!)^2}{{}_{2n+2}P_2}$(建部賢弘の公式)

$\arctan x = \sum_{n=0}^\infty \frac{(-1)^n}{2n+1} x^{2n+1}$を利用

$\frac{\pi}{4} = \arctan 1= \sum_{n=0}^\infty \frac{(-1)^n}{2n+1}$(ライプニッツの公式、マーダヴァとグレゴリーが先に発見)

$\pi = 6\arctan\frac{1}{\sqrt{3}} = \sqrt{12}\sum_{k=1}^\infty \frac{2(-1)^k}{(2k+1)3^k}$(マーダヴァの公式、シャープの公式)

$\frac{\pi}{2} = 2\arctan\frac{1}{\sqrt{2}} +\arctan\frac{1}{2\sqrt{2}} = 2\sqrt{2}(\sum_{n=1}^\infty \frac{(-1)^{n-1}}{2^n(2n-1)} +\sum_{n=1}^\infty \frac{(-1)^{n-1}}{2^{3n}(2n-1)})$

$\frac{\pi}{6} = 2\arctan\frac{1}{3\sqrt{3}} +\arctan\frac{1}{4\sqrt{3}} = \frac{\sqrt{3}}{36}(4\sum_{n=0}^\infty \frac{(-1)^n}{27^n(2n+1)} +3\sum_{n=0}^\infty \frac{(-1)^n}{48^n(2n+1)})$

$\frac{5\pi}{4} = 12\arctan\frac{1}{3} + 4\arctan\frac{1}{57} - \arctan\frac{1}{239}$

一般式

$\frac{\pi}{4} = P\arctan\frac{1}{2} - M\arctan\frac{1}{3} + L\arctan\frac{1}{5} + K\arctan\frac{1}{7} +$

$(N+K+L-2M+3P-5)\arctan\frac{1}{8} + (2N+M-P+2-L)\arctan\frac{1}{17}$

$- (2P-3-M+L+K-N)\arctan\frac{1}{58} - N\arctan\frac{1}{239}$

$\frac{\pi}{4} = (N+2)\arctan\frac{1}{2} - N\arctan\frac{1}{3}-(N+1)\arctan\frac{1}{N}$

(P,N,M,L,Kは正の実数)

無限和

$\frac{\pi}{4} = \sum_{n=1}^\infty \arctan\frac{1}{F_{2n+1}} = \arctan\frac{1}{2} + \arctan\frac{1}{5} + \arctan\frac{1}{13} + \cdots$

$\frac{\pi}{4} = \sum_{n=1}^\infty \arctan\frac{1}{n^2+n+1} = \arctan\frac{1}{3} + \arctan\frac{1}{7} + \arctan\frac{1}{13} + \cdots$

$\frac{\pi}{4} = \sum_{n=1}^\infty \arctan\frac{1}{2n^2} = \arctan\frac{1}{2} + \arctan\frac{1}{8} + \arctan\frac{1}{18} + \cdots$

おまけ(分母が連続する式)

$\frac{\pi}{4} = \arctan1 + \arctan\frac{1}{2} + \arctan\frac{1}{3} - \arctan\frac{1}{4} + 4\arctan\frac{1}{5} + \arctan\frac{1}{6} +$

$2\arctan\frac{1}{7} + 4\arctan\frac{1}{8} - 2\arctan\frac{1}{9} + \arctan\frac{1}{10} + \arctan\frac{1}{11} + \arctan\frac{1}{12} -$

$\arctan\frac{1}{13} + \arctan\frac{1}{14} - 2\arctan\frac{1}{15} + \arctan\frac{1}{16} + \arctan\frac{1}{17} + \arctan\frac{1}{50} +$

$\arctan\frac{1}{68} + \arctan\frac{1}{91} - \arctan\frac{1}{211} + \arctan\frac{1}{241}$

$\arctan x = \frac{1}{x}\sum_{n=0}^\infty \frac{(2n)!!}{(2n+1)!!} (\frac{x^2}{1+x^2})^{n+1}$を利用

$\frac{\pi}{4} = 5 \arctan\frac{1}{7} + 2 \arctan\frac{3}{79} =$

$35\sum_{n=0}^\infty \frac{(2n)!!}{(2n+1)!!}(\frac{2}{100})^{n+1} + \frac{158}{3}\sum_{n=0}^\infty \frac{(2n)!!}{(2n+1)!!}(\frac{144}{100000})^{n+1}$

(オイラーはこの公式を用いて、円周率を1時間で20桁まで求めた。)

$\frac{\pi}{4} = \arctan\frac{1}{7} + 2 \arctan\frac{1}{3} =$

$7\sum_{n=0}^\infty \frac{(2n)!!}{(2n+1)!!}(\frac{2}{100})^{n+1} + 6\sum_{n=0}^\infty \frac{(2n)!!}{(2n+1)!!}(\frac{1}{10})^{n+1}$