计算圆周率的反正切级数（马青类公式）

nyy

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

nyy

+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

nyy

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

lihpb00

不错不错

nyy

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

nyy

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！

nyy

+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

nyy

1. :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

1. +322[577] +76[682] +139[1393] +156[12943] +132[32807] +44[1049433]   == 1 * Pi/4
   
2. +122[319] +61[378] +115[557] +29[1068] +22[3458] +44[27493]   == 1 * Pi/4
   
3. +100[319] +127[378] +71[557] -15[1068] +66[2943] +44[478707]   == 1 * Pi/4
   
4. +337[307] -193[463] +151[4193] +305[4246] -122[39307] -83[390112]   == 1 * Pi/4
   
5. +183[268] +32[682] +95[1568] +44[4662] -166[12943] -51[32807]   == 1 * Pi/4
   
6. +183[268] +32[682] +95[1483] -7[9932] -122[12943] +51[29718]   == 1 * Pi/4
   
7. +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
\[+88\arctan{(\frac{1}{192})}+39\arctan{(\frac{1}{239})}+100\arctan{(\frac{1}{515})}-32\arctan{(\frac{1}{1068})}-56\arctan{(\frac{1}{173932})}=\frac{\pi}{4}\]

nyy

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：

\[
+166553022292\arctan{(\frac{1}{970522492753})}\\
+222363417479\arctan{(\frac{1}{989193552378})}\\
-134276698825\arctan{(\frac{1}{1096452832428})}\\
+168215423310\arctan{(\frac{1}{1280283860113})}\\
+75023059326\arctan{(\frac{1}{1341087111018})}\\
+136852193784\arctan{(\frac{1}{1689015353762})}\\
+217055842606\arctan{(\frac{1}{1822081215762})}\\
+103141369176\arctan{(\frac{1}{2184607268277})}\\
+28713480349\arctan{(\frac{1}{2278678014557})}\\
+221440571852\arctan{(\frac{1}{2635662131192})}\\
+184010343804\arctan{(\frac{1}{3165256360443})}\\
-130014434756\arctan{(\frac{1}{3385630462882})}\\
+30039704433\arctan{(\frac{1}{4426171412662})}\\
-125016355012\arctan{(\frac{1}{4963640229982})}\\
-268445832064\arctan{(\frac{1}{4972090102688})}\\
+80047317279\arctan{(\frac{1}{6306451059345})}\\
+229618316915\arctan{(\frac{1}{10221155603807})}\\
+30192504858\arctan{(\frac{1}{10305371319950})}\\
+18293883503\arctan{(\frac{1}{13688849577057})}\\
-44291036474\arctan{(\frac{1}{14483848717682})}\\
+29376832104\arctan{(\frac{1}{24632166555862})}\\
-139440534748\arctan{(\frac{1}{39537374317540})}\\
-59815251609\arctan{(\frac{1}{69971515635443})}\\
+62403552219\arctan{(\frac{1}{104225908824307})}\\
+59060238669\arctan{(\frac{1}{106851921608307})}\\
-169497968425\arctan{(\frac{1}{169838669284032})}\\
-238261971358\arctan{(\frac{1}{452493528674723})}\\
=\frac{\pi}{4}
\]

nyy

https://arxiv.org/pdf/2312.05413.pdf
这儿也有马青公式