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# [转载] 计算圆周率的反正切级数（马青类公式）

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Arctan relations for Pi

https://www.jjj.de/arctan/arctanpage.html

In a relation
M1*arctan(1/A1)+M2*arctan(1/A2)+...+Mj*arctan(1/Aj) == k*Pi/4
the left hand side is abbreviated as
M1[A1]+M2[A2]+...+Mj[Aj]
The term of least convergence is listed first. Relations of n arctan terms are in one file. The files are ordered according to the arguments, the "best" relation is first. When the first arguments coincide the next is used for ordering. An example (6-term relations):
+322[577] +76[682] +139[1393] +156[12943] +132[32807] +44[1049433]   == 1 * Pi/4
+122[319] +61[378] +115[557] +29[1068] +22[3458] +44[27493]   == 1 * Pi/4
+100[319] +127[378] +71[557] -15[1068] +66[2943] +44[478707]   == 1 * Pi/4
+337[307] -193[463] +151[4193] +305[4246] -122[39307] -83[390112]   == 1 * Pi/4
+183[268] +32[682] +95[1568] +44[4662] -166[12943] -51[32807]   == 1 * Pi/4
+183[268] +32[682] +95[1483] -7[9932] -122[12943] +51[29718]   == 1 * Pi/4
+29[268] +269[463] +154[2059] +122[2943] -186[9193] +71[390112]   == 1 * Pi/4
Each relation is followed by a list of primes of the form 4*k+1. These are obtained by factoring Ai^2+1 for each (inverse) argument Ai. An example (a 5-term relation):

+88[192] +39[239] +100[515] -32[1068] -56[173932]   == 1 * Pi/4
{5, 13, 73, 101}
We have
192^2+1 == 36865 == 5 73 101
239^2+1 == 57122 == 2 13 13 13 13
515^2+1 == 265226 == 2 13 101 101
1068^2+1 == 1140625 == 5 5 5 5 5 5 73
173932^2+1 == 30252340625 == 5 5 5 5 5 13 73 101 101

楼主| 发表于 2023-9-7 14:37:14 | 显示全部楼层
 +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]   == 1 * Pi/4      {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} 这个太极端了！ https://www.jjj.de/arctan/arctan-27term.txt

楼主| 发表于 2023-9-7 14:39:04 | 显示全部楼层
 2-term relations: +4[5] -1[239]   == 1 * Pi/4       {13} +2[3] +1[7]   == 1 * Pi/4       {5} +1[2] +1[3]   == 1 * Pi/4       {5} 3-term relations: +12[18] +8[57] -5[239]   == 1 * Pi/4       {5, 13} +8[10] -1[239] -4[515]   == 1 * Pi/4       {13, 101} +5[7] +4[68] +2[117]   == 1 * Pi/4       {5, 37} +5[7] +4[53] +2[4443]   == 1 * Pi/4       {5, 281} +5[7] +2[28] +2[443]   == 1 * Pi/4       {5, 157} 4-term relations: +44[57] +7[239] -12[682] +24[12943]   == 1 * Pi/4       {5, 13, 61} +20[57] +24[68] +12[117] -5[239]   == 1 * Pi/4       {5, 13, 37} +24[53] +20[57] -5[239] +12[4443]   == 1 * Pi/4       {5, 13, 281} +12[38] +20[57] +7[239] +24[268]   == 1 * Pi/4       {5, 13, 17} +24[29] -4[57] +7[239] -12[12238]   == 1 * Pi/4       {5, 13, 421} 5-term relations: +88[192] +39[239] +100[515] -32[1068] -56[173932]   == 1 * Pi/4       {5, 13, 73, 101} +88[172] +51[239] +32[682] +44[5357] +68[12943]   == 1 * Pi/4       {5, 13, 61, 97} +88[111] +7[239] -44[515] +32[682] +24[12943]   == 1 * Pi/4       {5, 13, 61, 101} +44[109] +95[239] -12[682] +24[12943] -44[6826318]   == 1 * Pi/4       {5, 13, 61, 457} +44[107] +44[122] +7[239] -12[682] +24[12943]   == 1 * Pi/4       {5, 13, 61, 229} 6-term relations: +322[577] +76[682] +139[1393] +156[12943] +132[32807] +44[1049433]   == 1 * Pi/4       {5, 13, 61, 89, 197} +122[319] +61[378] +115[557] +29[1068] +22[3458] +44[27493]   == 1 * Pi/4       {5, 17, 41, 73, 181} +100[319] +127[378] +71[557] -15[1068] +66[2943] +44[478707]   == 1 * Pi/4       {5, 13, 17, 41, 73} +337[307] -193[463] +151[4193] +305[4246] -122[39307] -83[390112]   == 1 * Pi/4       {5, 13, 17, 29, 97} +183[268] +32[682] +95[1568] +44[4662] -166[12943] -51[32807]   == 1 * Pi/4       {5, 13, 17, 61, 89} 7-term relations: +1587[2852] +295[4193] ... -708[390112]   == 1 * Pi/4       {5, 13, 17, 29, 97, 433} +327[1958] +481[2059] ... +398[390112]   == 1 * Pi/4       {5, 13, 17, 41, 97, 349} +1074[1568] +657[4662] ... +398[390112]   == 1 * Pi/4       {5, 13, 17, 61, 89, 97} +1106[1143] -330[4193] ... +398[390112]   == 1 * Pi/4       {5, 13, 17, 29, 53, 97} +481[1084] +295[4193] ... -227[390112]   == 1 * Pi/4       {5, 13, 17, 29, 97, 409} 8-term relations: +2192[5357] +2097[5507] ... -708[1115618]   == 1 * Pi/4       {5, 13, 29, 37, 61, 97, 337} +1484[5118] +708[6072] ... -398[619858]   == 1 * Pi/4       {5, 13, 17, 29, 53, 269, 373} +1882[5118] +1106[6072] ... +398[390112]   == 1 * Pi/4       {5, 13, 17, 41, 53, 97, 373} +2805[4662] +1257[5357] ... +1074[1493208]   == 1 * Pi/4       {5, 13, 17, 61, 89, 97, 233} +2363[4557] +1218[5507] ... +481[27872057]   == 1 * Pi/4       {5, 13, 17, 29, 37, 97, 449} 9-term relations: +3286[34208] +9852[39307] ... +776[976283]   == 1 * Pi/4       {5, 13, 17, 29, 41, 53, 97, 269} +6832[25368] +4062[34208] ... -1882[619858]   == 1 * Pi/4       {5, 13, 17, 29, 53, 97, 269, 433} +9012[18543] +6896[39307] ... +4062[27872057]   == 1 * Pi/4       {5, 13, 17, 29, 37, 97, 433, 449} +9852[17298] +5546[34208] ... +8300[1460857]   == 1 * Pi/4       {5, 13, 17, 29, 53, 109, 157, 269} +5280[15789] +4838[34208] ... -1882[619858]   == 1 * Pi/4       {5, 13, 17, 29, 53, 97, 269, 281} 10-term relations: +1106[54193] -30569[78629] ... +23407[201229582]   == -1 * Pi/4       {5, 13, 17, 41, 53, 73, 97, 101, 157} +13301[54193] +19560[66347] ... -5280[193788912]   == 1 * Pi/4       {5, 13, 17, 37, 41, 53, 73, 101, 157} +27764[51693] +18979[138724] ... +3581[227661182]   == 1 * Pi/4       {5, 13, 17, 29, 53, 109, 233, 457, 569} +50539[51387] +1555[114669] ... +25433[24208144]   == 1 * Pi/4       {5, 13, 17, 29, 37, 53, 61, 89, 97} +24891[51387] +26988[83270] ... +776[657922943]   == 1 * Pi/4       {5, 13, 17, 29, 37, 53, 89, 109, 233} 11-term relations: +36462[390112] +135908[485298] ... -43938[2189376182]   == 1 * Pi/4       {5, 13, 17, 29, 37, 53, 61, 89, 97, 101} +52094[333367] +29861[390112] ... +43938[103224943]   == 1 * Pi/4       {5, 13, 17, 29, 37, 53, 61, 89, 97, 241} +37616[232643] +29861[275807] ... +6056[33853568]   == 1 * Pi/4       {5, 13, 17, 29, 41, 53, 61, 157, 197, 269} +59324[219602] +46743[275807] ... +27764[21072618]   == 1 * Pi/4       {5, 13, 17, 29, 53, 109, 137, 269, 457, 593} +102486[219602] +46743[275807] ... -43162[227661182]   == 1 * Pi/4       {5, 13, 17, 29, 53, 109, 137, 233, 269, 457} 12-term relations: +893758[1049433] +655711[1264557] ... -432616[2189376182]   == 2 * Pi/4       {5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 197} +619249[683982] -211951[1984933] ... -216308[2189376182]   == 1 * Pi/4       {5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 281} +446879[683982] +172370[1635786] ... -216308[2189376182]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101} +483341[683982] -36462[1059193] ... -216308[2189376182]   == 1 * Pi/4       {5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 181} +290735[683982] -68439[783568] ... -59551[653507178]   == 1 * Pi/4       {5, 13, 17, 29, 37, 61, 101, 109, 181, 193, 337} 13-term relations: +1126917[3449051] +1337518[4417548] ... -216308[2189376182]   == 1 * Pi/4       {5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 181, 281} +1241486[2478328] +292729[4180652] ... -407298[7830729512]   == 1 * Pi/4       {5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 281, 733} +1860735[2478328] -114569[3449051] ... -623606[2189376182]   == 1 * Pi/4       {5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 241, 281} +1241486[2478328] +831200[3014557] ... -216308[2189376182]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 281} +286458[2357607] -1377426[2715193] ... +519636[7959681215]   == -1 * Pi/4       {5, 13, 17, 41, 53, 97, 101, 109, 149, 193, 277, 601} 14-term relations: +446879[6826318] +5624457[8082212] ... +483341[17249711432]   == 1 * Pi/4       {5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 181, 269, 457} +1126917[6826318] -7198253[8082212] ... -3591352[17249711432]   == -1 * Pi/4       {5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 181, 281, 457} +2701787[6682866] +1721624[6826318] ... +224134[30446482737]   == 1 * Pi/4       {5, 13, 17, 41, 53, 61, 89, 109, 113, 137, 241, 269, 457} +1821154[6656382] +2369262[8296072] ... +4799[3069221943]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 61, 113, 137, 149, 229, 449, 557} +3801953[6367252] -1532570[8082212] ... +1337518[4006581229]   == 1 * Pi/4       {5, 13, 17, 29, 37, 53, 61, 73, 89, 97, 101, 181, 281} 15-term relations: +5034126[20942043] +1546003[22709274] ... +1337518[250645741818]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 181, 337, 389} +5345097[18975991] +6293190[22709274] ... +4944419[7804016832]   == 1 * Pi/4       {5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 109, 233, 277, 557} +980346[18975991] +6580129[22709274] ... +2603331[18986886768]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 181, 389, 457} +5752395[18975991] +1808080[22709274] ... -2815282[164432798314]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 109, 389, 541} +6371644[18975991] +1188831[22709274] ... +2603331[18986886768]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 181, 389, 461} 16-term relations: +14215326[53141564] +6973645[54610269] ... +8735690[34840696582]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 101, 109, 233, 241, 389, 569} +17294544[52153699] +27205340[55906808] ... -13226263[134520516108]   == 1 * Pi/4       {5, 13, 17, 29, 37, 53, 89, 97, 101, 109, 193, 229, 233, 557, 757} +12552413[51853693] +33848374[62660863] ... +11582317[442812063749]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 109, 149, 461, 617} +8897246[49480595] +16223408[52352552] ... +6195674[121409547033]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 73, 97, 101, 113, 229, 409, 433, 709} +6453528[47927722] +11661213[49480595] ... +6115274[512223806648]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 73, 97, 101, 113, 197, 229, 409, 433} 17-term relations: +12872838[201229582] +27205340[203420807] ... +35839320[134520516108]   == 1 * Pi/4       {5, 13, 17, 29, 37, 53, 89, 97, 101, 109, 113, 197, 229, 233, 557, 757} +73667294[160007778] -19737150[167207057] ... +73757780[25793659208]   == 1 * Pi/4       {5, 13, 17, 29, 41, 53, 61, 73, 109, 113, 137, 149, 157, 181, 409, 421} +39580760[159358932] +16166691[160007778] ... -32961758[617280798753]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 73, 97, 109, 113, 149, 157, 193, 229, 449, 557} +20700907[159358932] -14295479[160007778] ... -53662765[617280798753]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 73, 97, 109, 113, 149, 157, 229, 293, 449, 557} +60126052[159358932] -4378601[160007778] ... -60592901[617280798753]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 73, 97, 109, 113, 149, 157, 193, 229, 457, 557} 18-term relations: +2859494[299252491] -41068896[321390012] ... -89623108[18004873694818]   == -1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 73, 97, 109, 113, 149, 157, 181, 193, 337, 409} +79635304[299252491] -41619921[303690197] ... -82571160[202198944907]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 61, 73, 89, 97, 109, 113, 149, 157, 193, 293, 449} +128948755[295651068] +46365822[374107307] ... +54020630[25793659208]   == 1 * Pi/4       {5, 13, 17, 29, 41, 53, 61, 73, 109, 113, 137, 149, 157, 181, 193, 409, 421} +24101193[284862638] +173878369[321390012] ... +116515842[250645741818]   == 2 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 97, 101, 109, 181, 193, 197, 277, 337, 409} +61004459[284862638] +33797796[313467682] ... +32726322[232367315757]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 97, 109, 137, 157, 197, 229, 241, 277, 337, 409} 19-term relations: +270619381[778401733] -138919506[1012047353] ... +146407224[30038155625330]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 73, 89, 97, 109, 113, 149, 157, 193, 257, 293, 449} +59529729[609849572] +79674619[642183094] ... +74693424[2292939666693]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 101, 113, 137, 157, 181, 233, 313, 461} +81426443[599832943] +121593960[619364243] ... +10896101[1029114662298]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 61, 73, 89, 97, 113, 157, 173, 181, 197, 233, 269, 277} +110095319[595484173] -107544826[924984650] ... -46539715[193100304493]   == -1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 113, 157, 173, 193, 241, 257, 613} +50408000[575512868] +56219270[619364243] ... +92203391[45055404168]   == 1 * Pi/4       {5, 13, 17, 37, 41, 53, 61, 73, 89, 113, 137, 157, 181, 197, 269, 277, 293, 373} 20-term relations: +807092487[2674664693] +479094776[2701984943] ... +214188292[564340076432]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 113, 149, 157, 193, 277, 313, 421, 509} +87218705[2572149874] +110260180[2635549633] ... -54384134[199921633110818]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 113, 149, 193, 257, 281, 349, 613} +128838741[2420845318] -48554212[2674664693] ... -703647950[69971515635443]   == -1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 137, 157, 173, 193, 257, 277, 337, 709} +476582424[2189376182] +330885384[2209555165] ... +1290385324[13474294213307]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 109, 149, 173, 257, 269, 313, 457, 617} +209679377[2131759027] +832386402[2189376182] ... -120586758[199921633110818]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 113, 173, 193, 257, 281, 349, 613} 21-term relations: +598245178[5513160193] -115804626[7622130953] ... -1521437626[38057255532937]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 149, 173, 233, 269, 281, 313, 349} +1636945012[5475957057] -2733315404[5513160193] ... +1772787486[38057255532937]   == -1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 149, 173, 233, 257, 269, 313, 349} +22036970[5299020696] -2300654420[5368767682] ... -430112898[69971515635443]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 137, 157, 173, 233, 269, 281, 349, 409} +2277397987[4549886677] -2588087820[6746573033] ... +883258705[686308367425978]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 109, 113, 137, 149, 173, 257, 293, 449, 457} +30168848[4504597011] -517851720[4732978887] ... -1055052705[69971515635443]   == -1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 137, 149, 173, 233, 257, 457, 521} 22-term relations: +2242198001[17249711432] +1907212074[18986886768] ... -2418616720[69971515635443]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 137, 149, 173, 181, 257, 449, 457, 617} +1687553954[13795202057] +2140036840[17337001791] ... +2839615695[69971515635443]   == -1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 173, 257, 313, 449, 457} +4294694239[12988236682] +8975120280[13795202057] ... -4584031221[32028841498519]   == -1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 109, 113, 149, 157, 193, 241, 269, 313, 449, 457} +2255035212[12957904393] -1719738754[17249711432] ... -9538773149[69971515635443]   == -1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 137, 149, 173, 181, 257, 269, 457, 617} +8852363052[12957904393] -6310889300[15234751332] ... -4662202233[1981522054472108]   == -2 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 137, 149, 173, 181, 257, 457, 617, 757} 23-term relations: +5667127453[58482499557] +15129381[59168467669] ... +8164579419[21881002771243]   == 2 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 97, 109, 113, 157, 173, 193, 197, 233, 241, 269, 313, 449, 521} +38687548853[57821838732] -12882562179[58482499557] ... -28435786819[38690844808533]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 97, 109, 113, 137, 157, 193, 197, 233, 241, 269, 313, 409, 449} +2245366820[30446482737] +10430675125[34319970459] ... +8433093551[47782248283866]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 97, 109, 113, 137, 157, 181, 193, 233, 241, 269, 281, 313, 449} +10000836358[29072499217] +16644416156[31147706551] ... -10655494736[240926005152903]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 97, 101, 109, 113, 137, 157, 173, 197, 233, 269, 313, 397, 521} +16212175144[28382409317] -4443575442[58482499557] ... -3747481728[21881002771243]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 97, 109, 113, 157, 173, 193, 197, 233, 241, 269, 313, 433, 449} 24-term relations: +25755712641[102416588812] +31142028402[102428655030] ... +32555698322[12970464759326963]   == 2 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 149, 173, 181, 193, 229, 257, 277, 281, 313, 433, 673} +53112874273[88125874307] +8412707264[93432718569] ... -21214686561[74295753693510382]   == 3 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 109, 113, 149, 173, 193, 197, 257, 269, 293, 313, 397, 509} +42395129953[76306498765] +53149822100[85486957057] ... +23815390395[1367017926441055]   == 3 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 173, 181, 193, 229, 277, 337, 353, 409} +6169334688[73798390077] +7410876424[76306498765] ... -682488502[4280680038152162]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 109, 113, 137, 173, 181, 193, 229, 277, 313, 337, 353, 421} +15758305932[73276714818] +6933500078[76612121243] ... -18697733393[295440851934557]   == -1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 149, 157, 173, 233, 241, 269, 293, 313, 397, 677} 25-term relations: +167174919693[160422360532] +28100366064[186974489515] ... +23981185267[3386696477716649]   == 4 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 181, 193, 229, 241, 277, 293, 577, 601} +63104267593[160422360532] -49239532881[164788939557] ... -26715804188[3386696477716649]   == -2 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 109, 113, 137, 149, 173, 181, 193, 229, 241, 277, 293, 601, 617} +40267708165[147207444057] +15351299076[166846193428] ... +48204000632[17251727577487177]   == 1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 157, 181, 229, 233, 241, 269, 313, 353, 577} +28523690053[147207444057] +7718539660[163365074403] ... -48204000632[17251727577487177]   == -1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 157, 181, 197, 229, 233, 241, 269, 313, 353, 577} +51859431398[138873731225] -30366643323[176888171806] ... -4338698676[169838669284032]   == -1 * Pi/4       {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 173, 193, 229, 257, 277, 317, 337, 601} 26-term relations: +152014292229[392943720343] +60325885083[476704288228] ... -103240793335[1367017926441055]   == 1 * Pi/4       {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] ... +48204000632[17251727577487177]   == 1 * Pi/4       {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] ... -115480230498[3386696477716649]   == -3 * Pi/4       {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] ... -21670383406[6805164953551432]   == -2 * Pi/4       {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] ... -247705754204[2799978903689557]   == 8 * Pi/4       {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-term relations: +166553022292[970522492753] +222363417479[989193552378] ... -238261971358[452493528674723]   == 1 * Pi/4       {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} https://www.jjj.de/arctan/best-arctan-relations.txt

 不错不错

楼主| 发表于 2023-9-8 09:24:34 | 显示全部楼层
 本帖最后由 nyy 于 2023-9-8 14:36 编辑 PAIRS INCORPORATING 5 DISTINCT COTANGENT VALUES Compound measure = 2.38268  (includes 5 DCVs) 1.58604  STØRMER (1896)    44[57]+7[239]-12[682]+24[12943]   Eliminated:{A}[11] 1.81316  HCL (04Jun95) 3[1]=44[18]-23[239]+8[682]-16[12943] 写成LaTeX，如下： $\frac{Pi}{4}=44\arctan{(\frac{1}{57})}+7\arctan{(\frac{1}{239})}-12\arctan{(\frac{1}{682})}+24\arctan{(\frac{1}{12943})}$ $\frac{3Pi}{4}=44\arctan{\frac{1}{18}}-23\arctan{\frac{1}{239}}+8\arctan{\frac{1}{682}}-16\arctan{\frac{1}{12943}}$ Top of frame http://www.machination.eclipse.co.uk/FSChecking.html

楼主| 发表于 2023-9-8 11:02:55 | 显示全部楼层
 https://www.jjj.de/arctan/arndt-arctan-2006.pdf.gz 如何发现这个反正切级数的呢？这个文章里有说明！

楼主| 发表于 2023-9-8 11:10:58 | 显示全部楼层
 本帖最后由 nyy 于 2023-9-8 11:38 编辑 +36462[390112] +135908[485298] +274509[683982] -39581[1984933] +178477[2478328] -114569[3449051] -146571[18975991] +61914[22709274] -69044[24208144] -89431[201229582] -43938[2189376182] == 1 * Pi/4 代表着： $+36462\arctan{\frac{1}{390112}}+135908\arctan{\frac{1}{485298}}\\ +274509\arctan{\frac{1}{683982}}-39581\arctan{\frac{1}{1984933}}\\ +178477\arctan{\frac{1}{2478328}}-114569\arctan{\frac{1}{3449051}}\\ -146571\arctan{\frac{1}{18975991}}+61914\arctan{\frac{1}{22709274}}\\ -69044\arctan{\frac{1}{24208144}}-89431\arctan{\frac{1}{201229582}}\\ -43938\arctan{\frac{1}{2189376182}}==1*\frac{Pi}{4}$ 我用LaTeX重新表达一下，这下更容易理解与明白

楼主| 发表于 2023-9-8 14:44:01 | 显示全部楼层
 本帖最后由 nyy 于 2023-9-8 14:52 编辑 :s/$$$\d\+$$$/\\arctan{(\\frac{1}{\1})}/gec 复制代码 这个是vim中替换的代码，我还是忍不住要把集中的几个搞成LaTeX +322[577] +76[682] +139[1393] +156[12943] +132[32807] +44[1049433]   == 1 * Pi/4 +122[319] +61[378] +115[557] +29[1068] +22[3458] +44[27493]   == 1 * Pi/4 +100[319] +127[378] +71[557] -15[1068] +66[2943] +44[478707]   == 1 * Pi/4 +337[307] -193[463] +151[4193] +305[4246] -122[39307] -83[390112]   == 1 * Pi/4 +183[268] +32[682] +95[1568] +44[4662] -166[12943] -51[32807]   == 1 * Pi/4 +183[268] +32[682] +95[1483] -7[9932] -122[12943] +51[29718]   == 1 * Pi/4 +29[268] +269[463] +154[2059] +122[2943] -186[9193] +71[390112]   == 1 * Pi/4 复制代码 这7行的LaTeX，分别表示 $+322\arctan{(\frac{1}{577})}+76\arctan{(\frac{1}{682})}+139\arctan{(\frac{1}{1393})}+156\arctan{(\frac{1}{12943})}+132\arctan{(\frac{1}{32807})}+44\arctan{(\frac{1}{1049433})}=\frac{\pi}{4}$ $+122\arctan{(\frac{1}{319})}+61\arctan{(\frac{1}{378})}+115\arctan{(\frac{1}{557})}+29\arctan{(\frac{1}{1068})}+22\arctan{(\frac{1}{3458})}+44\arctan{(\frac{1}{27493})}=\frac{\pi}{4}$ $+100\arctan{(\frac{1}{319})}+127\arctan{(\frac{1}{378})}+71\arctan{(\frac{1}{557})}-15\arctan{(\frac{1}{1068})}+66\arctan{(\frac{1}{2943})}+44\arctan{(\frac{1}{478707})}=\frac{\pi}{4}$ $+337\arctan{(\frac{1}{307})}-193\arctan{(\frac{1}{463})}+151\arctan{(\frac{1}{4193})}+305\arctan{(\frac{1}{4246})}-122\arctan{(\frac{1}{39307})}-83\arctan{(\frac{1}{390112})}=\frac{\pi}{4}$ $+183\arctan{(\frac{1}{268})}+32\arctan{(\frac{1}{682})}+95\arctan{(\frac{1}{1568})}+44\arctan{(\frac{1}{4662})}-166\arctan{(\frac{1}{12943})}-51\arctan{(\frac{1}{32807})}=\frac{\pi}{4}$ $+183\arctan{(\frac{1}{268})}+32\arctan{(\frac{1}{682})}+95\arctan{(\frac{1}{1483})}-7\arctan{(\frac{1}{9932})}-122\arctan{(\frac{1}{12943})}+51\arctan{(\frac{1}{29718})}=\frac{\pi}{4}$ $+29\arctan{(\frac{1}{268})}+269\arctan{(\frac{1}{463})}+154\arctan{(\frac{1}{2059})}+122\arctan{(\frac{1}{2943})}-186\arctan{(\frac{1}{9193})}+71\arctan{(\frac{1}{390112})}=\frac{\pi}{4}$ 表示 +88[192]+39[239]+100[515]-32[1068]-56[173932]==1*Pi/4 $+88\arctan{(\frac{1}{192})}+39\arctan{(\frac{1}{239})}+100\arctan{(\frac{1}{515})}-32\arctan{(\frac{1}{1068})}-56\arctan{(\frac{1}{173932})}=\frac{\pi}{4}$

### 点评

楼主| 发表于 2023-9-11 11:04:44 | 显示全部楼层
 nyy 发表于 2023-9-7 14:37 +166553022292[970522492753] +222363417479[989193552378] -134276698825[1096452832428] +168215423310[1 ... 把二楼公式LaTeX化，如下： $+166553022292\arctan{(\frac{1}{970522492753})}\\ +222363417479\arctan{(\frac{1}{989193552378})}\\ -134276698825\arctan{(\frac{1}{1096452832428})}\\ +168215423310\arctan{(\frac{1}{1280283860113})}\\ +75023059326\arctan{(\frac{1}{1341087111018})}\\ +136852193784\arctan{(\frac{1}{1689015353762})}\\ +217055842606\arctan{(\frac{1}{1822081215762})}\\ +103141369176\arctan{(\frac{1}{2184607268277})}\\ +28713480349\arctan{(\frac{1}{2278678014557})}\\ +221440571852\arctan{(\frac{1}{2635662131192})}\\ +184010343804\arctan{(\frac{1}{3165256360443})}\\ -130014434756\arctan{(\frac{1}{3385630462882})}\\ +30039704433\arctan{(\frac{1}{4426171412662})}\\ -125016355012\arctan{(\frac{1}{4963640229982})}\\ -268445832064\arctan{(\frac{1}{4972090102688})}\\ +80047317279\arctan{(\frac{1}{6306451059345})}\\ +229618316915\arctan{(\frac{1}{10221155603807})}\\ +30192504858\arctan{(\frac{1}{10305371319950})}\\ +18293883503\arctan{(\frac{1}{13688849577057})}\\ -44291036474\arctan{(\frac{1}{14483848717682})}\\ +29376832104\arctan{(\frac{1}{24632166555862})}\\ -139440534748\arctan{(\frac{1}{39537374317540})}\\ -59815251609\arctan{(\frac{1}{69971515635443})}\\ +62403552219\arctan{(\frac{1}{104225908824307})}\\ +59060238669\arctan{(\frac{1}{106851921608307})}\\ -169497968425\arctan{(\frac{1}{169838669284032})}\\ -238261971358\arctan{(\frac{1}{452493528674723})}\\ =\frac{\pi}{4}$

### 点评

楼主| 发表于 2023-12-27 13:43:55 | 显示全部楼层
 https://arxiv.org/pdf/2312.05413.pdf 这儿也有马青公式

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