FANDOM


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
51[239] +132[268] +44[599] +32[682] -44[1118] -20[12943] 5, 13, 17, 61, 173
83[239] +100[268] +32[557] +12[1068] -44[15449] -44[51693] 5, 13, 17, 73, 569
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
83[239] +100[268] +32[557] -32[1068] +44[1568] +44[4662] 5, 13, 17, 73, 89
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
77[239] +138[268] -26[463] +32[5604] -64[51693] -6[390112] 5, 13, 17, 97, 569
61[239] +122[268] +22[463] +64[2682] +32[8827] -22[390112] 5, 13, 17, 97, 677
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
71[239] +100[268] +56[395] -64[2943] -32[8783] -44[37107] 5, 13, 17, 41, 353
39[239] +100[268] +88[395] +64[2682] +32[8827] -44[37107] 5, 13, 17, 353, 677
7[239] +132[268] +88[395] +32[682] -64[12943] -44[37107] 5, 13, 17, 61, 353
39[239] +100[268] +88[395] +32[557] -32[1068] -44[37107] 5, 13, 17, 73, 353
115[239] +144[268] -76[378] -108[2943] +44[14318] +44[27493] 5, 13, 17, 41, 181
93[239] +100[268] +12[378] -42[2943] +44[7535] -22[131168] 5, 13, 17, 41, 241
115[239] +100[268] -32[378] -20[2943] +88[4662] +44[11981] 5, 13, 17, 41, 89
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
115[239] +144[268] -32[378] -44[421] -44[1985] -64[2943] 5, 13, 17, 41, 401
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
39[239] +68[268] +120[307] -12[795] -56[13307] -44[14256] 5, 13, 17, 29, 641
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
327[1958] +481[2059] +625[2928] +449[2943] +298[4052] -513[9193] +398[390112] 5, 13, 17, 41, 97, 349
1074[1568] +657[4662] +183[5357] -779[12943] -32[17923] -449[32807] +398[390112] 5, 13, 17, 61, 89, 97
1106[1143] -330[4193] -513[4246] +481[34208] +215[39307] -144[44179] +398[390112] 5, 13, 17, 29, 53, 97
481[1084] +295[4193] +737[4246] +625[6687] +215[39307] -144[235318] -227[390112] 5, 13, 17, 29, 97, 409
764[1068] +127[3323] +442[20457] +215[37057] +188[83247] +171[173932] +481[409557] 5, 13, 29, 73, 101, 149
581[993] +171[2072] +127[2633] +24[2943] +227[5593] +342[16432] -100[6826318] 5, 13, 37, 41, 257, 457
327[905] +298[1057] +183[2059] +151[2943] -29[3957] +83[9193] +100[390112] 5, 13, 17, 41, 97, 109
144[905] +481[948] +337[3957] +151[4193] -32[4246] +215[39307] -83[390112] 5, 13, 17, 29, 97, 109
537[882] +581[2917] -322[9466] +371[12943] -171[19703] -266[32807] -76[157318] 5, 13, 29, 37, 61, 89
625[837] +183[2059] -176[2943] +83[9193] -29[247057] +100[390112] +327[740943] 5, 13, 17, 41, 97, 317
625[829] -144[2059] +151[4193] +112[9193] +449[9872] +327[39307] -227[390112] 5, 13, 17, 29, 41, 97
625[811] +183[2059] -176[2943] -244[9193] +356[29193] +100[390112] -327[1366849] 5, 13, 17, 41, 97, 617
481[798] +625[3141] +144[3323] -330[4193] +112[4246] -266[39307] -227[390112] 5, 13, 17, 29, 97, 101
481[795] +295[4193] +256[4246] +625[13307] +144[14256] -266[39307] -227[390112] 5, 13, 17, 29, 97, 641
83[788] +342[837] +7[2943] +83[5503] +71[247057] -100[352618] +144[740943] 5, 13, 17, 41, 233, 317
210[771] +242[882] +190[1168] +95[1393] +66[6948] -76[45368] -132[2947340] 5, 29, 37, 197, 277, 577
359[757] +122[813] +200[1068] -105[3112] +83[12238] +44[58911] +132[3139557] 5, 13, 29, 73, 157, 421
537[746] +44[5357] +259[5507] +156[12943] -171[17923] +227[32807] +139[157318] 5, 13, 37, 61, 89, 97
537[746] +44[2621] +371[12943] +215[14942] +271[32807] -171[83071] -76[157318] 5, 13, 37, 61, 89, 193
537[746] +44[1992] +171[9222] +371[12943] -215[16693] +271[32807] +139[157318] 5, 13, 37, 61, 89, 241
322[746] +215[1393] +474[2547] +156[12943] +56[32807] -76[157318] +44[1049433] 5, 13, 37, 61, 89, 197
266[746] +215[1067] +327[1568] +144[4662] -115[13918] -100[27493] -32[157318] 5, 13, 17, 37, 89, 181
266[746] +271[882] +315[2621] -56[14942] -171[19703] +371[25943] -76[157318] 5, 13, 29, 37, 89, 193
266[746] +215[882] +427[1568] -371[6107] -315[18543] -171[19703] -132[157318] 5, 13, 17, 29, 37, 89
398[743] +139[1393] +322[2153] +76[8307] -166[12943] +132[32807] +44[1049433] 5, 13, 61, 89, 181, 197
266[743] +271[882] +315[2917] +171[8307] -227[9466] +200[12943] +95[1059193] 5, 13, 29, 37, 61, 181
14 446879[6826318] +5624457[8082212] +582656[12477035] -106990[18975991] +1013473[22709274] -824070[24208144] +1534215[53798468] -2636805[63199427] -880919[201229582] -1573796[231373438] -851058[306903943] -1106947[2189376182] +39581[2539791558] +483341[17249711432] 5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 181, 269, 457
1126917[6826318] -7198253[8082212] -1573796[8188568] -2156452[12477035] +1720367[18975991] +560323[22709274] +2397866[24208144] -39581[25010458] +4210601[63199427] +4028511[201229582] +2464435[306903943] -1534215[462333568] +2680743[2189376182] -3591352[17249711432]=-[1] 5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 181, 281, 457
2701787[6682866] +1721624[6826318] +3446790[7465443] -2719182[7626068] +542267[9639557] -152627[21072618] -1563678[72079977] +1625409[103224943] -1716488[130357318] -562730[144152533] -2004760[2340083537] -3132751[2539791558] -2489653[18221678207] +224134[30446482737] 5, 13, 17, 41, 53, 61, 89, 109, 113, 137, 241, 269, 457
1821154[6656382] +2369262[8296072] +2506408[9689961] -243296[10292025] -2048571[14495807] +1288329[24789357] +1028644[27872057] +1997881[40939124] +128993[46401003] -591043[159358932] -396321[215266693] -768959[227327735] +1363317[1379896643] +4799[3069221943] 5, 13, 17, 29, 37, 41, 61, 113, 137, 149, 229, 449, 557
3801953[6367252] -1532570[8082212] +446879[8188568] +3704571[12477035] +1870985[18975991] -770924[22709274] -144032[24208144] -1297937[25010458] +2055787[63199427] -1126917[193788912] -1774677[201229582] +407298[462333568] +1121210[2189376182] +1337518[4006581229] 5, 13, 17, 29, 37, 53, 61, 73, 89, 97, 101, 181, 281
3434531[6225244] +1337518[7042807] -1238498[8071877] +2218844[18975991] +2427329[22709274] +1607565[24185182] -29463[24208144] -3123560[52630807] -1860735[54610269] -718269[103224943] +428637[115453918] +804327[201229582] +601007[909140573] +328248[2189376182] 5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 241, 389
2097013[6225244] +1337518[6797757] +718269[8735943] +1599595[18975991] +1089811[22709274] +1706585[24185182] -29463[24208144] -523217[54610269] -1885062[104029218] -639136[115453918] +86058[201229582] +1418296[909140573] -739525[2189376182] -1786042[3712239557] 5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 389, 569
138601[6225244] +3295930[6367252] -350847[7042807] +1628795[8071877] -648449[18975991] -439964[22709274] +2087539[24208144] +1688365[26063363] +970096[103224943] -428637[193788912] -1890596[201229582] -681807[266981905] +172370[1770638022] +1587976[2189376182] 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 241
619249[6208047] -4053780[6225244] -2937113[18975991] -718269[21638297] -2427329[22709274] -369067[24185182] -688806[24208144] +3123560[52630807] +1860735[54610269] +2147379[115453918] -1423576[201229582] +637491[909140573] -1046517[2189376182] -1956767[3508373380]=-1[1] 5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 197, 389
1573796[5033696] +407298[6826318] +2595608[8082212] +582656[12845247] -186152[18975991] +973892[22709274] +710145[24208144] -1142171[63199427] +911978[168623905] -337844[201229582] -307983[306903943] -1146528[2189376182] +522922[17249711432] -39581[66492889557] 5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 109, 181, 457
2937113[5033696] -1821154[6225244] +2318572[8296072] +890639[9639557] +2448822[12137341] -1152440[19696179] -540353[22709274] +1666164[24208144] -1106947[36175498] -1990985[40971432] +1334806[168623905] +2163355[284862638] -619249[313467682] -1268359[29546599818] 5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 109, 137, 409
1631996[5033696] -58200[6225244] +1305117[6826318] +555618[8296072] +271390[12845247] +585479[19696179] +223205[21072618] +1222601[22709274] +1666164[24208144] +396044[44300557] +2676681[168623905] +400401[284862638] +649110[318942476] +457837[66492889557] 5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 109, 137, 457
2960373[5033696] -602928[6225244] -29861[6622698] +1106947[6682866] +1539749[7042807] -291662[8071877] -1890596[8960203] +1450632[22709274] +2147261[24208144] +1100346[103224943] +358109[168623905] -130250[231373438] -904716[284862638] -1990985[18221678207] 5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 109, 241, 269
427268[5033696] +1955212[6225244] +2509845[6553113] +432802[12645777] -82516[22709274] +533191[23458141] +439403[24185182] -34997[24208144] -1457794[27005643] -808684[54610269] -723803[168623905] +1086269[284862638] +619249[367829623] +890639[909140573] 5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 109, 337, 389
820819[5033696] -3379042[6225244] -3851720[6553113] +984427[6682866] +451236[7042807] -1699323[8071877] +100389[12645777] +439964[22709274] +1734320[24208144] +1990985[27005643] +890639[103224943] +1532487[168623905] -1086269[284862638] -1961124[367829623]=-[1] 5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 109, 241, 337
2504016[5033696] -2224874[5835807] +3059357[8519197] -497418[18975991] +3510032[22709274] +4136924[24208144] +1802617[168623905] +900359[201229582] +1393969[225823278] -2632172[227661182] -579373[657922943] +2536140[934981432] -2133199[2189376182] +1605920[7804016832] 5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 109, 233, 557
1880705[5033696] +747649[5530088] +1333456[6826318] -834188[6898468] -190509[11131086] +981535[12845247] +1189905[22709274] -342201[23747457] +2383952[26431400] +2365415[168623905] +1457794[2420845318] +618175[19785169368] +1413856[66492889557] +90896[120563046313] 5, 13, 17, 29, 37, 61, 73, 89, 97, 109, 173, 277, 457
1111236[4980443] +1076378[5033696] +1457794[6208047] +749499[6826318] +363360[12845247] -960376[15265047] -121831[21638297] +1280801[22709274] -930515[40748057] -925084[119742462] +1565445[168623905] +1176389[176717682] +653467[284862638] +544376[66492889557] 5, 13, 17, 29, 37, 41, 61, 89, 97, 109, 197, 349, 457
320986[4980443] +1076378[5033696] +1539749[5982670] +1358733[6225244] -731065[9548768] -71065[15265047] +1099785[22709274] +1374502[24208144] +227198[119742462] -649110[159031137] +59185[162252407] +1606196[168623905] -96032[284862638] +100389[804120217] 5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 109, 157, 349
1746166[4956193] +3394655[6858314] +311266[8071877] -483341[8188568] -790147[18975991] +681163[22709274] +1308055[24208144] -1509888[25010458] -1860735[92985095] +930220[103224943] -2590328[201229582] +2460078[2189376182] -1222949[2322848793] -1337518[7830729512] 5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 241, 281, 733
649110[4956193] +1241486[5982670] +1060470[6225244] +1403965[6412637] -357448[8188568] -432802[9548768] +1278072[11918668] +369897[22709274] +66569[24208144] -678434[24660332] +1005313[25010458] +789073[92985095] -1273715[1545736093] -96032[7830729512] 5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 157, 281, 733
2448822[4735368] +161412[4980443] +326879[5033696] +627668[6225244] +1699323[8296072] +890639[9639557] -1829573[15265047] -991028[19696179] -540353[22709274] -324821[24208144] -649110[37611943] -722626[119742462] +66447[168623905] +894996[284862638] 5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 109, 137, 349
222694[4641970] -3064634[5842386] +44531[5982670] -345384[7042807] -2136924[9548768] -1332494[26063363] +400402[34713882] +37106[103224943] -384106[244653118] -754401[261221282] -1009411[2189376182] +1599793[2701984943] +696231[5475957057] -261801[193100304493]=-[1] 5, 13, 17, 29, 41, 53, 61, 73, 89, 101, 157, 233, 241
1241486[4640657] +579668[7220803] +2193045[9582318] +1158528[18975991] +1395319[19754557] +394224[22709274] +1544333[24208144] +446879[77861243] -2235614[193788912] +1136637[201229582] -1262530[345311195] -138601[1454097393] -354909[2189376182] -1281067[29546599818] 5, 13, 17, 29, 37, 53, 61, 73, 89, 97, 101, 373, 409
1573796[4640657] -446879[6826318] -8772049[8082212] +1613377[11788873] -2235614[12477035] -1466806[18975991] -973892[22709274] +2437447[24208144] +4210601[63199427] +2494296[201229582] +2464435[306903943] +2680743[2189376182] -2017556[17249711432] -1534215[29546599818]=-[1] 5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 181, 409, 457
1241486[4640657] +1606215[4916855] +1262530[6367252] +350847[9582318] +264770[18975991] -632323[22709274] +517786[24208144] -1026547[30808747] -446879[34713882] -762188[193788912] -125893[201229582] -718269[1454097393] +91970[2189376182] +192359[29546599818] 5, 13, 17, 29, 37, 53, 61, 73, 89, 97, 101, 233, 409
1337518[4417548] +2097013[8188568] +619249[11918668] -1834936[18975991] +1808080[22709274] +3902618[24185182] -29463[24208144] +2196033[25010458] -407298[52630807] -1241486[54610269] -2906874[115453918] +86058[201229582] -538471[909140573] -1050496[2189376182] 5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 281, 389
1337518[4417548] +2479984[8082212] +236278[8188568] +1029535[12923243] -593450[18975991] +566594[22709274] +1224299[24185182] -241414[24208144] +1166498[25010458] -619249[63199427] -745142[201229582] -622237[462333568] +1126917[909140573] -216308[2189376182] 5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 181, 281, 389
1337518[4417548] +3606901[8082212] +236278[8188568] +1029535[12477035] +533467[18975991] -560323[22709274] -1270949[24208144] +39581[25010458] -619249[63199427] -1126917[144309885] -1774677[201229582] +407298[462333568] -1126917[593897943] -216308[2189376182] 5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 181, 257, 281
2464435[4417548] +4733818[8082212] -2017556[8188568] +1029535[12477035] -1720367[18975991] -560323[22709274] -2397866[24208144] +1166498[25010458] -1746166[63199427] +1126917[92985095] -2901594[201229582] +1534215[462333568] -216308[2189376182] +1126917[7830729512] 5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 181, 281, 733
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
2464435[4417548] +3606901[8082212] -890639[8188568] -97382[12477035] -593450[18975991] -1126917[20942043] +566594[22709274] -1270949[24208144] +1166498[25010458] -1746166[63199427] -1774677[201229582] +407298[462333568] -1343225[2189376182] -1126917[250645741818] 5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 181, 281, 337
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
2269383[4417548] +114569[6225244] -4555658[6858314] -3606901[7042807] +2072686[8071877] -1262825[8188568] -1860735[11918668] -1364667[18975991] -566594[22709274] +29463[24208144] +172370[25010458] +292729[103224943] -378787[201229582] +509037[2189376182]=-1[1] 5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 241, 281
921918[4417548] +8302[6225244] +3191301[6367252] -271390[8188568] +203649[11918668] -708019[18975991] +150994[22709274] +2746238[24208144] -172370[25010458] +1657086[60803114] -1118615[193788912] -1571028[201229582] +1440778[2189376182] +1241486[5142102426318] 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 281
96032[4417548] +2987652[5033696] -890639[8188568] +1648784[11280443] -1212699[18975991] +834188[19005382] +1808080[22709274] +1504752[24208144] +39581[25010458] +2365415[168623905] -745142[201229582] -619249[546365045] +402941[2189376182] +1860735[164432798314] 5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 109, 281, 541
1337518[4417548] +1126917[5033696] +2479984[8082212] -890639[8188568] +1029535[12477035] -593450[18975991] +566594[22709274] -144032[24208144] +39581[25010458] -619249[63199427] +1126917[168623905] -1774677[201229582] +407298[462333568] -216308[2189376182] 5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 109, 181, 281
930220[4417548] +1453437[4956193] +1534215[5033696] -483341[8188568] +1241486[17931732] -1427638[18975991] +973892[22709274] +670564[24208144] -579668[25010458] -619249[92985095] +292729[168623905] -125893[201229582] -4357[2189376182] -407298[7830729512] 5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 109, 281, 733
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
2775701[4394207] +2040533[4417548] -1110313[6225244] -974405[8188568] +1322264[11918668] -2945249[18975991] +2237230[19177543] -967621[22709274] -29463[24208144] -172370[25010458] -1944501[57882807] +86058[201229582] +1317907[2189376182] +703015[12924735728] 5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 281, 593
619249[4286937] +930515[4956193] +954547[5033696] +930220[6826318] +367717[12845247] +834188[17931732] +25504[18975991] +877565[22709274] +90896[24208144] +815651[168623905] +804327[201229582] -527279[2189376182] -522922[2740120011] +483341[66492889557] 5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 109, 457, 733
1403965[4218932] +1957043[5926518] -66447[5982670] +1060470[6225244] -2087293[6444407] -357448[8188568] +216308[9548768] +3395226[11804243] -29861[11918668] -2236256[22709274] +100787[24208144] +3269870[24660332] -302620[25010458] -649110[199223608] 5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 157, 281, 317
207379[4218932] +697286[4773557] +334673[5982670] +1448423[6315768] +1324229[6909739] +955057[9548768] +384106[40939124] +317425[60033932] -717429[160007778] +940123[212087312] -660712[361632045] +1361845[2971354082] -417416[5271470807] +839138[14033378718] 5, 13, 17, 29, 41, 53, 61, 73, 113, 149, 157, 181, 229
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
176727[4218932] -1324229[4352888] -697286[4773557] -718779[5982670] -508300[6315768] -955057[9548768] -940123[13850847] -701531[60033932] +384106[63199427] -606800[160007778] +276606[361632045] +346490[2971354082] +33310[5271470807] +485091[14033378718]=-[1] 5, 13, 17, 29, 41, 53, 61, 73, 101, 113, 149, 157, 181
136453[4218932] +768212[4327682] -11817[5982670] +1670719[6208047] +1150177[9548768] +869197[9897688] +1832529[15961432] +249750[60033932] +275564[85639497] -1656312[110586432] -1377099[160007778] -205680[361632045] +1185628[648111693] +455032[1735194643] 5, 13, 17, 29, 41, 53, 61, 73, 149, 157, 197, 709, 757
2157526[4180652] -411360[4956193] +1678645[8071877] +884038[8188568] -130250[8538694] -563052[12477035] -253414[22709274] +1438305[24208144] +1080440[25010458] +296791[92985095] +627668[103224943] -1367379[462333568] -934577[1144849597] -1337518[7830729512] 5, 13, 17, 29, 37, 53, 61, 89, 97, 181, 241, 281, 733
579668[4180652] +1337518[4640657] +1166498[4956193] +350847[6456617] -593450[18975991] +1884767[21638297] -288638[22709274] +613818[24313557] -125893[201229582] +1397069[2189376182] +682862[5332311473] +2262602[6884660047] -446879[8181920807] +1332382[29546599818] 5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 197, 409, 733
1534215[4180652] +2171706[4417548] +1453437[4956193] -1724827[8188568] -2669124[18975991] -560323[22709274] -863651[24208144] +954547[25010458] +914966[92985095] -1367379[201229582] +1241486[487989368] +1241486[1333514268] +1237129[2189376182] -407298[7830729512] 5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 281, 701, 733
1534215[4180652] +930220[4417548] +211951[4956193] -483341[8188568] +1241486[9768407] +1241486[18107068] -186152[18975991] -560323[22709274] +377835[24208144] +954547[25010458] +914966[92985095] -125893[201229582] -4357[2189376182] -407298[7830729512] 5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 281, 677, 733
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
1241486[4055901] +1534215[4180652] -311266[4417548] +211951[4956193] -483341[8188568] +1241486[11804243] +1055334[18975991] -560323[22709274] +377835[24208144] +954547[25010458] +914966[92985095] -125893[201229582] -4357[2189376182] -407298[7830729512] 5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 281, 317, 733
658823[4031104] +754855[4218932] +582663[5982670] +1060470[6225244] +291662[8188568] +216308[9548768] +649110[9839162] +619249[11918668] -139963[13879557] +509860[22709274] +100787[24208144] -125356[24660332] -302620[25010458] -130250[528115843] 5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 157, 281, 421
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
980346[18975991] +6580129[22709274] +1087336[24185182] +4742586[24208144] +5391298[53828768] +1573796[54610269] +3434531[63199427] +4984000[100656977] -4032441[115453918] +4238858[201229582] -1337518[306903943] -9463320[909140573] -2176063[2189376182] -4053780[17249711432] +2603331[18986886768] 5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 181, 389, 457
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
4797848[18975991] +8206580[21638297] -52655[22709274] -8357742[24185182] +10370162[24208144] +4053780[119742462] +6246825[168623905] +1549469[176717682] -4364751[180567850] +804327[201229582] +3434531[240109693] -3474112[909140573] +3837472[2189376182] +3817502[3508373380] -1860735[1442060060693] 5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 109, 197, 349, 389
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
3434531[15060457] +5752395[18975991] -619249[20983750] +1808080[22709274] +3663352[24185182] -648712[24208144] +4772049[50693817] -718269[51853693] -4461078[52630807] +2193045[54610269] +6538204[115453918] -2815282[193788912] -533191[201229582] +2656794[909140573] +3003284[2189376182] 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 389, 461
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}

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