论坛 › 数学欣赏 › 计算圆周率的反正切级数（马青类公式）

nyy 发表于 2023-9-7 14:35:57

计算圆周率的反正切级数（马青类公式）

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+M2+...+Mj
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 +76 +139 +156 +132 +44   == 1 * Pi/4
+122 +61 +115 +29 +22 +44   == 1 * Pi/4
+100 +127 +71 -15 +66 +44   == 1 * Pi/4
+337 -193 +151 +305 -122 -83   == 1 * Pi/4
+183 +32 +95 +44 -166 -51   == 1 * Pi/4
+183 +32 +95 -7 -122 +51   == 1 * Pi/4
+29 +269 +154 +122 -186 +71   == 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 +39 +100 -32 -56   == 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 发表于 2023-9-7 14:37:14

+166553022292 +222363417479 -134276698825 +168215423310 +75023059326 +136852193784 +217055842606 +103141369176 +28713480349 +221440571852 +184010343804 -130014434756 +30039704433 -125016355012 -268445832064 +80047317279 +229618316915 +30192504858 +18293883503 -44291036474 +29376832104 -139440534748 -59815251609 +62403552219 +59060238669 -169497968425 -238261971358   == 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 发表于 2023-9-7 14:39:04

2-term relations:
+4 -1   == 1 * Pi/4
      {13}


+2 +1   == 1 * Pi/4
      {5}


+1 +1   == 1 * Pi/4
      {5}




3-term relations:
+12 +8 -5   == 1 * Pi/4
      {5, 13}


+8 -1 -4   == 1 * Pi/4
      {13, 101}


+5 +4 +2   == 1 * Pi/4
      {5, 37}


+5 +4 +2   == 1 * Pi/4
      {5, 281}


+5 +2 +2   == 1 * Pi/4
      {5, 157}




4-term relations:
+44 +7 -12 +24   == 1 * Pi/4
      {5, 13, 61}


+20 +24 +12 -5   == 1 * Pi/4
      {5, 13, 37}


+24 +20 -5 +12   == 1 * Pi/4
      {5, 13, 281}


+12 +20 +7 +24   == 1 * Pi/4
      {5, 13, 17}


+24 -4 +7 -12   == 1 * Pi/4
      {5, 13, 421}




5-term relations:
+88 +39 +100 -32 -56   == 1 * Pi/4
      {5, 13, 73, 101}


+88 +51 +32 +44 +68   == 1 * Pi/4
      {5, 13, 61, 97}


+88 +7 -44 +32 +24   == 1 * Pi/4
      {5, 13, 61, 101}


+44 +95 -12 +24 -44   == 1 * Pi/4
      {5, 13, 61, 457}


+44 +44 +7 -12 +24   == 1 * Pi/4
      {5, 13, 61, 229}




6-term relations:
+322 +76 +139 +156 +132 +44   == 1 * Pi/4
      {5, 13, 61, 89, 197}


+122 +61 +115 +29 +22 +44   == 1 * Pi/4
      {5, 17, 41, 73, 181}


+100 +127 +71 -15 +66 +44   == 1 * Pi/4
      {5, 13, 17, 41, 73}


+337 -193 +151 +305 -122 -83   == 1 * Pi/4
      {5, 13, 17, 29, 97}


+183 +32 +95 +44 -166 -51   == 1 * Pi/4
      {5, 13, 17, 61, 89}




7-term relations:
+1587 +295 ... -708   == 1 * Pi/4
      {5, 13, 17, 29, 97, 433}

+327 +481 ... +398   == 1 * Pi/4
      {5, 13, 17, 41, 97, 349}

+1074 +657 ... +398   == 1 * Pi/4
      {5, 13, 17, 61, 89, 97}

+1106 -330 ... +398   == 1 * Pi/4
      {5, 13, 17, 29, 53, 97}

+481 +295 ... -227   == 1 * Pi/4
      {5, 13, 17, 29, 97, 409}



8-term relations:
+2192 +2097 ... -708   == 1 * Pi/4
      {5, 13, 29, 37, 61, 97, 337}

+1484 +708 ... -398   == 1 * Pi/4
      {5, 13, 17, 29, 53, 269, 373}

+1882 +1106 ... +398   == 1 * Pi/4
      {5, 13, 17, 41, 53, 97, 373}

+2805 +1257 ... +1074   == 1 * Pi/4
      {5, 13, 17, 61, 89, 97, 233}

+2363 +1218 ... +481   == 1 * Pi/4
      {5, 13, 17, 29, 37, 97, 449}



9-term relations:
+3286 +9852 ... +776   == 1 * Pi/4
      {5, 13, 17, 29, 41, 53, 97, 269}

+6832 +4062 ... -1882   == 1 * Pi/4
      {5, 13, 17, 29, 53, 97, 269, 433}

+9012 +6896 ... +4062   == 1 * Pi/4
      {5, 13, 17, 29, 37, 97, 433, 449}

+9852 +5546 ... +8300   == 1 * Pi/4
      {5, 13, 17, 29, 53, 109, 157, 269}

+5280 +4838 ... -1882   == 1 * Pi/4
      {5, 13, 17, 29, 53, 97, 269, 281}



10-term relations:
+1106 -30569 ... +23407   == -1 * Pi/4
      {5, 13, 17, 41, 53, 73, 97, 101, 157}

+13301 +19560 ... -5280   == 1 * Pi/4
      {5, 13, 17, 37, 41, 53, 73, 101, 157}

+27764 +18979 ... +3581   == 1 * Pi/4
      {5, 13, 17, 29, 53, 109, 233, 457, 569}

+50539 +1555 ... +25433   == 1 * Pi/4
      {5, 13, 17, 29, 37, 53, 61, 89, 97}

+24891 +26988 ... +776   == 1 * Pi/4
      {5, 13, 17, 29, 37, 53, 89, 109, 233}



11-term relations:
+36462 +135908 ... -43938   == 1 * Pi/4
      {5, 13, 17, 29, 37, 53, 61, 89, 97, 101}

+52094 +29861 ... +43938   == 1 * Pi/4
      {5, 13, 17, 29, 37, 53, 61, 89, 97, 241}

+37616 +29861 ... +6056   == 1 * Pi/4
      {5, 13, 17, 29, 41, 53, 61, 157, 197, 269}

+59324 +46743 ... +27764   == 1 * Pi/4
      {5, 13, 17, 29, 53, 109, 137, 269, 457, 593}

+102486 +46743 ... -43162   == 1 * Pi/4
      {5, 13, 17, 29, 53, 109, 137, 233, 269, 457}



12-term relations:
+893758 +655711 ... -432616   == 2 * Pi/4
      {5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 197}

+619249 -211951 ... -216308   == 1 * Pi/4
      {5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 281}

+446879 +172370 ... -216308   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101}

+483341 -36462 ... -216308   == 1 * Pi/4
      {5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 181}

+290735 -68439 ... -59551   == 1 * Pi/4
      {5, 13, 17, 29, 37, 61, 101, 109, 181, 193, 337}



13-term relations:
+1126917 +1337518 ... -216308   == 1 * Pi/4
      {5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 181, 281}

+1241486 +292729 ... -407298   == 1 * Pi/4
      {5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 281, 733}

+1860735 -114569 ... -623606   == 1 * Pi/4
      {5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 241, 281}

+1241486 +831200 ... -216308   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 281}

+286458 -1377426 ... +519636   == -1 * Pi/4
      {5, 13, 17, 41, 53, 97, 101, 109, 149, 193, 277, 601}



14-term relations:
+446879 +5624457 ... +483341   == 1 * Pi/4
      {5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 181, 269, 457}

+1126917 -7198253 ... -3591352   == -1 * Pi/4
      {5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 181, 281, 457}

+2701787 +1721624 ... +224134   == 1 * Pi/4
      {5, 13, 17, 41, 53, 61, 89, 109, 113, 137, 241, 269, 457}

+1821154 +2369262 ... +4799   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 61, 113, 137, 149, 229, 449, 557}

+3801953 -1532570 ... +1337518   == 1 * Pi/4
      {5, 13, 17, 29, 37, 53, 61, 73, 89, 97, 101, 181, 281}



15-term relations:
+5034126 +1546003 ... +1337518   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 181, 337, 389}

+5345097 +6293190 ... +4944419   == 1 * Pi/4
      {5, 13, 17, 29, 37, 53, 61, 89, 97, 101, 109, 233, 277, 557}

+980346 +6580129 ... +2603331   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 181, 389, 457}

+5752395 +1808080 ... -2815282   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 109, 389, 541}

+6371644 +1188831 ... +2603331   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 181, 389, 461}



16-term relations:
+14215326 +6973645 ... +8735690   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 61, 73, 101, 109, 233, 241, 389, 569}

+17294544 +27205340 ... -13226263   == 1 * Pi/4
      {5, 13, 17, 29, 37, 53, 89, 97, 101, 109, 193, 229, 233, 557, 757}

+12552413 +33848374 ... +11582317   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 109, 149, 461, 617}

+8897246 +16223408 ... +6195674   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 73, 97, 101, 113, 229, 409, 433, 709}

+6453528 +11661213 ... +6115274   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 73, 97, 101, 113, 197, 229, 409, 433}



17-term relations:
+12872838 +27205340 ... +35839320   == 1 * Pi/4
      {5, 13, 17, 29, 37, 53, 89, 97, 101, 109, 113, 197, 229, 233, 557, 757}

+73667294 -19737150 ... +73757780   == 1 * Pi/4
      {5, 13, 17, 29, 41, 53, 61, 73, 109, 113, 137, 149, 157, 181, 409, 421}

+39580760 +16166691 ... -32961758   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 73, 97, 109, 113, 149, 157, 193, 229, 449, 557}

+20700907 -14295479 ... -53662765   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 73, 97, 109, 113, 149, 157, 229, 293, 449, 557}

+60126052 -4378601 ... -60592901   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 73, 97, 109, 113, 149, 157, 193, 229, 457, 557}



18-term relations:
+2859494 -41068896 ... -89623108   == -1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 73, 97, 109, 113, 149, 157, 181, 193, 337, 409}

+79635304 -41619921 ... -82571160   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 61, 73, 89, 97, 109, 113, 149, 157, 193, 293, 449}

+128948755 +46365822 ... +54020630   == 1 * Pi/4
      {5, 13, 17, 29, 41, 53, 61, 73, 109, 113, 137, 149, 157, 181, 193, 409, 421}

+24101193 +173878369 ... +116515842   == 2 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 61, 97, 101, 109, 181, 193, 197, 277, 337, 409}

+61004459 +33797796 ... +32726322   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 97, 109, 137, 157, 197, 229, 241, 277, 337, 409}



19-term relations:
+270619381 -138919506 ... +146407224   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 73, 89, 97, 109, 113, 149, 157, 193, 257, 293, 449}

+59529729 +79674619 ... +74693424   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 101, 113, 137, 157, 181, 233, 313, 461}

+81426443 +121593960 ... +10896101   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 61, 73, 89, 97, 113, 157, 173, 181, 197, 233, 269, 277}

+110095319 -107544826 ... -46539715   == -1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 113, 157, 173, 193, 241, 257, 613}

+50408000 +56219270 ... +92203391   == 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 +479094776 ... +214188292   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 113, 149, 157, 193, 277, 313, 421, 509}

+87218705 +110260180 ... -54384134   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 113, 149, 193, 257, 281, 349, 613}

+128838741 -48554212 ... -703647950   == -1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 137, 157, 173, 193, 257, 277, 337, 709}

+476582424 +330885384 ... +1290385324   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 61, 89, 97, 101, 109, 149, 173, 257, 269, 313, 457, 617}

+209679377 +832386402 ... -120586758   == 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 -115804626 ... -1521437626   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 149, 173, 233, 269, 281, 313, 349}

+1636945012 -2733315404 ... +1772787486   == -1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 149, 173, 233, 257, 269, 313, 349}

+22036970 -2300654420 ... -430112898   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 137, 157, 173, 233, 269, 281, 349, 409}

+2277397987 -2588087820 ... +883258705   == 1 * Pi/4
      {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 109, 113, 137, 149, 173, 257, 293, 449, 457}

+30168848 -517851720 ... -1055052705   == -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 +1907212074 ... -2418616720   == 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 +2140036840 ... +2839615695   == -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 +8975120280 ... -4584031221   == -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 -1719738754 ... -9538773149   == -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 -6310889300 ... -4662202233   == -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 +15129381 ... +8164579419   == 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 -12882562179 ... -28435786819   == 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 +10430675125 ... +8433093551   == 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 +16644416156 ... -10655494736   == 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 -4443575442 ... -3747481728   == 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 +31142028402 ... +32555698322   == 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 +8412707264 ... -21214686561   == 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 +53149822100 ... +23815390395   == 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 +7410876424 ... -682488502   == 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 +6933500078 ... -18697733393   == -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 +28100366064 ... +23981185267   == 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 -49239532881 ... -26715804188   == -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 +15351299076 ... +48204000632   == 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 +7718539660 ... -48204000632   == -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 -30366643323 ... -4338698676   == -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 +60325885083 ... -103240793335   == 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 -112112075621 ... +48204000632   == 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 -182115180498 ... -115480230498   == -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 -239339538150 ... -21670383406   == -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 +755595651182 ... -247705754204   == 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 +222363417479 ... -238261971358   == 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 发表于 2023-9-7 22:05:11

不错不错

nyy 发表于 2023-9-8 09:24:34

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)
1.58604STØRMER (1896)
   44+7-12+24
Eliminated:{A}
1.81316HCL (04Jun95)
3=44-23+8-16

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 发表于 2023-9-8 11:02:55

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 发表于 2023-9-8 11:10:58

本帖最后由 nyy 于 2023-9-8 11:38 编辑

+36462 +135908 +274509 -39581
+178477 -114569 -146571 +61914
-69044 -89431 -43938 == 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 发表于 2023-9-8 14:44:01

本帖最后由 nyy 于 2023-9-8 14:52 编辑

:s/\[\(\d\+\)\]/\\arctan{(\\frac{1}{\1})}/gec


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 +76 +139 +156 +132 +44   == 1 * Pi/4
+122 +61 +115 +29 +22 +44   == 1 * Pi/4
+100 +127 +71 -15 +66 +44   == 1 * Pi/4
+337 -193 +151 +305 -122 -83   == 1 * Pi/4
+183 +32 +95 +44 -166 -51   == 1 * Pi/4
+183 +32 +95 -7 -122 +51   == 1 * Pi/4
+29 +269 +154 +122 -186 +71   == 1 * Pi/4


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+39+100-32-56==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 发表于 2023-9-11 11:04:44

nyy 发表于 2023-9-7 14:37
+166553022292 +222363417479 -134276698825 +168215423310

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 发表于 2023-12-27 13:43:55

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

查看完整版本: 计算圆周率的反正切级数（马青类公式）