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根据楼上 wanye 提供的相关资料及计算共有116组解(按照hyperellratpoints(f(u),v),1000000)计算结果)
[序号,\([u,v,D,x,y,z]\)]
[1, [-9/20, 1000/47, 495260031/441800, 414560/422481, 95800/422481, 217519/422481]],
[2, [-9/20, 1000/47, -495260031/441800, -632671960/1679142729, -1670617271/1679142729, 50237800/1679142729]],
[3, [-9/20, 5728/215, 16723774473/9245000, 2767624/2813001, 673865/2813001, 1390400/2813001]],
[4, [-9/20, 5728/215, -16723774473/9245000, -5821981400/15434547801, -15355831360/15434547801, 140976551/15434547801]],
[5, [-9/20, -1041/320, 2126704839/40960000, 1670617271/1679142729, 632671960/1679142729, -50237800/1679142729]],
[6, [-9/20, -1041/320, -2126704839/40960000, -95800/422481, -414560/422481, -217519/422481]],
[7, [-9/20, -1425/412, 3894577617/67897600, 15355831360/15434547801, 5821981400/15434547801, -140976551/15434547801]],
[8, [-9/20, -1425/412, -3894577617/67897600, -673865/2813001, -2767624/2813001, -1390400/2813001]],
[9, [-9/20, -4209/3500, 6507898503/700000000, 414560/422481, 95800/422481, -217519/422481]],
[10, [-9/20, -4209/3500, -6507898503/700000000, 7592431981391/29999857938609, -22495595284040/29999857938609, -27239791692640/29999857938609]],
[11, [-9/20, 30080/6007, 195619772649/7216809800, 22495595284040/29999857938609, -7592431981391/29999857938609, 27239791692640/29999857938609]],
[12, [-9/20, 30080/6007, -195619772649/7216809800, -95800/422481, -414560/422481, 217519/422481]],
[13, [-9/20, 34225/6692, 75863359911/2559020800, 440804942580160/573646321871961, -130064300991400/573646321871961, 514818101299289/573646321871961]],
[14, [-9/20, 34225/6692, -75863359911/2559020800, -673865/2813001, -2767624/2813001, 1390400/2813001]],
[15, [-9/20, -41952/33865, 2280363347913/229367645000, 2767624/2813001, 673865/2813001, -1390400/2813001]],
[16, [-9/20, -41952/33865, -2280363347913/229367645000, 130064300991400/573646321871961, -440804942580160/573646321871961, -514818101299289/573646321871961]],
[17, [-29/12, 1865/132, 4566509815/2509056, 894416022327/2051764828361, -125777308440/2051764828361, 2032977944240/2051764828361]],
[18, [-29/12, 1865/132, -4566509815/2509056, -8332208/8707481, -5507880/8707481, -1705575/8707481]],
[19, [-29/12, 6280/1359, 15222896105/132975432, -7908038161032/45556888578449, -27546142170735/45556888578449, 43940127884360/45556888578449]],
[20, [-29/12, 6280/1359, -15222896105/132975432, -8332208/8707481, -5507880/8707481, 1705575/8707481]],
[21, [-29/12, -30768/57253, 1190554344625/33715604664, 5507880/8707481, 8332208/8707481, 1705575/8707481]],
[22, [-29/12, -30768/57253, -1190554344625/33715604664, 125777308440/2051764828361, -894416022327/2051764828361, -2032977944240/2051764828361]],
[23, [-29/12, -3333/107368, 27781785391625/1660015789056, 5507880/8707481, 8332208/8707481, -1705575/8707481]],
[24, [-29/12, -3333/107368, -27781785391625/1660015789056, 27546142170735/45556888578449, 7908038161032/45556888578449, -43940127884360/45556888578449]],
[25, [-93/80, -400/37, 2960618799/4380800, 11289040/12197457, 5870000/12197457, 8282543/12197457]],
[26, [-93/80, -400/37, -2960618799/4380800, -1237796960/1787882337, -1662997663/1787882337, -686398000/1787882337]],
[27, [-93/80, -2433/920, 372167198439/5416960000, 1662997663/1787882337, 1237796960/1787882337, 686398000/1787882337]],
[28, [-93/80, -2433/920, -372167198439/5416960000, -5870000/12197457, -11289040/12197457, -8282543/12197457]],
[29, [-93/80, -84237/359800, 66222454729143/118359808000000, 11289040/12197457, 5870000/12197457, -8282543/12197457]],
[30, [-93/80, -84237/359800, -66222454729143/118359808000000, 15876595946759369395903/17503689286309573964097, 7188470920864810763360/17503689286309573964097, -12896301483090810351440/17503689286309573964097]],
[31, [-400/37, -93/80, 2960618799/4380800, 686398000/1787882337, 1662997663/1787882337, 1237796960/1787882337]],
[32, [-400/37, -93/80, -2960618799/4380800, -8282543/12197457, -5870000/12197457, -11289040/12197457]],
[33, [-400/37, -2433/920, 985189443879/579360800, 5870000/12197457, 8282543/12197457, 11289040/12197457]],
[34, [-400/37, -2433/920, -985189443879/579360800, -1662997663/1787882337, -686398000/1787882337, -1237796960/1787882337]],
[35, [-136/133, 201/4, 1416600375/141512, 14173720/16003017, 4479031/16003017, 12552200/16003017]],
[36, [-136/133, 201/4, -1416600375/141512, -108644015/156646737, -146627384/156646737, -27450160/156646737]],
[37, [-136/133, -1005/568, 102051891015/2853447968, 146627384/156646737, 108644015/156646737, 27450160/156646737]],
[38, [-136/133, -1005/568, -102051891015/2853447968, -4479031/16003017, -14173720/16003017, -12552200/16003017]],
[39, [201/4, -136/133, 1416600375/141512, 27450160/156646737, 146627384/156646737, 108644015/156646737]],
[40, [201/4, -136/133, -1416600375/141512, -12552200/16003017, -4479031/16003017, -14173720/16003017]],
[41, [201/4, -1005/568, 87999215295/5161984, 4479031/16003017, 12552200/16003017, 14173720/16003017]],
[42, [201/4, -1005/568, -87999215295/5161984, -146627384/156646737, -27450160/156646737, -108644015/156646737]],
[43, [-5/8, -1617/200, 708019737/2560000, 630662624/638523249, 275156240/638523249, 219076465/638523249]],
[44, [-5/8, -1617/200, -708019737/2560000, -260052385/589845921, -582665296/589845921, -186668000/589845921]],
[45, [-5/8, -477/692, 64335705/30647296, 18796760/20615673, 2682440/20615673, -15365639/20615673]],
[46, [-5/8, -477/692, -64335705/30647296, 2448718655/3393603777, -664793200/3393603777, -3134081336/3393603777]],
[47, [-5/8, 20824/2003, 32269755735/128384288, 18796760/20615673, 2682440/20615673, 15365639/20615673]],
[48, [-5/8, 20824/2003, -32269755735/128384288, -2480452675600/5062297699257, -4987588419655/5062297699257, 502038853976/5062297699257]],
[49, [-5/8, -34272/4885, 164968598721/763623200, 582665296/589845921, 260052385/589845921, 186668000/589845921]],
[50, [-5/8, -34272/4885, -164968598721/763623200, -275156240/638523249, -630662624/638523249, -219076465/638523249]],
[51, [-5/8, 36696/8687, 2141459895/344978144, 664793200/3393603777, -2448718655/3393603777, 3134081336/3393603777]],
[52, [-5/8, 36696/8687, -2141459895/344978144, -2682440/20615673, -18796760/20615673, 15365639/20615673]],
[53, [-5/8, 398113/66200, 16290521488377/280476160000, 47886740272114976/62940516903410601, -8813425670440240/62940516903410601, 56827813308111785/62940516903410601]],
[54, [-5/8, 398113/66200, -16290521488377/280476160000, -260052385/589845921, -582665296/589845921, 186668000/589845921]],
[55, [-5/8, -124529/68084, 7518853086255/296667587584, 4987588419655/5062297699257, 2480452675600/5062297699257, -502038853976/5062297699257]],
[56, [-5/8, -124529/68084, -7518853086255/296667587584, -2682440/20615673, -18796760/20615673, -15365639/20615673]],
[57, [-5/8, -176752/157345, 9325236881511/792238368800, 630662624/638523249, 275156240/638523249, -219076465/638523249]],
[58, [-5/8, -176752/157345, -9325236881511/792238368800, 3579087147375440/20249506709579721, -14890026433468471/20249506709579721, -18565945114216720/20249506709579721]],
[59, [233/60, -216285/23504, 675018780820577/1988776857600, -7813353720/12558554489, 4707813440/12558554489, 11988496761/12558554489]],
[60, [233/60, -216285/23504, -675018780820577/1988776857600, -335981923744570504065/375075545025537358721, 188195571677171463096/375075545025537358721, 275897431444390465240/375075545025537358721]],
[61, [233/60, 7584/54605, 42947654622593/5367070845000, -343651286746207896/481334894209428521, 438980913824794665/481334894209428521, -225712385669145920/481334894209428521]],
[62, [233/60, 7584/54605, -42947654622593/5367070845000, -7813353720/12558554489, 4707813440/12558554489, -11988496761/12558554489]],
[63, [-56/165, -383021/380940, 997084931509393/1975381798005000, 1047978087905/1367141947873, -408600530760/1367141947873, -1226022682752/1367141947873]],
[64, [-56/165, -383021/380940, -997084931509393/1975381798005000, 163180699054891578792/228746036963039501833, -84616109521023161865/228746036963039501833, -210878774189729581880/228746036963039501833]],
[65, [-125/92, -936/5281, 514986989385/118026082952, 23449050222680/25866132798297, 18776929334105/25866132798297, -12035933588696/25866132798297]],
[66, [-125/92, -936/5281, -514986989385/118026082952, 52289667920/64244765937, 17111129720/64244765937, -55479193841/64244765937]],
[67, [-361/540, 1861/240, 2085707016991/16796160000, 122055375/145087793, 1841160/145087793, 121952168/145087793]],
[68, [-361/540, 1861/240, -2085707016991/16796160000, -535914713672/1059621884297, -1041572957760/1059621884297, 187577183625/1059621884297]],
[69, [-361/540, -7800/5509, 80577403041031/4424896009800, 1041572957760/1059621884297, 535914713672/1059621884297, -187577183625/1059621884297]],
[70, [-361/540, -7800/5509, -80577403041031/4424896009800, -1841160/145087793, -122055375/145087793, -121952168/145087793]],
[71, [-817/660, -1581/1520, 23439166132879/1006410240000, 9585769407872803575/10816708329115215113, 8510180374729994520/10816708329115215113, 985735303963754488/10816708329115215113]],
[72, [-817/660, -1581/1520, -23439166132879/1006410240000, -2164632/44310257, -31669120/44310257, -41084175/44310257]],
[73, [-817/660, 21792/5035, 148223121024127/5521496805000, -2164632/44310257, -31669120/44310257, 41084175/44310257]],
[74, [-817/660, 21792/5035, -148223121024127/5521496805000, -1199828498161126807800/1671674986261410994097, -1534990269771364822095/1671674986261410994097, 38585919318751039616/98333822721259470241]],
[75, [-865/592, -14177/20156, 2838307044130425/142381024251904, 1553556440/1871713857, 1593513080/1871713857, 92622401/1871713857]],
[76, [-865/592, -14177/20156, -2838307044130425/142381024251904, 208032601069058735/1592672455342770513, -851144034922098880/1592672455342770513, -1559028675188874616/1592672455342770513]],
[77, [-865/592, -230472/438737, 527721585216584535/33730434870574208, 1553556440/1871713857, 1593513080/1871713857, -92622401/1871713857]],
[78, [-865/592, -230472/438737, -527721585216584535/33730434870574208, 179164925544119666072000/587020625514136613276553, -222787467202130880567415/587020625514136613276553, -582653975191641098286104/587020625514136613276553]],
[79, [553/80, -33400/19537, 260117908400151/1221421980800, 24743080/5179020201, 3971389576/5179020201, 4657804375/5179020201]],
[80, [553/80, -33400/19537, -260117908400151/1221421980800, -5352683902805120/5380742305932201, 1554532675059625/5380742305932201, -1841841620201576/5380742305932201]],
[81, [553/80, -294473/635180, 207130360353696711/2582103247360000, -1554532675059625/5380742305932201, 5352683902805120/5380742305932201, 1841841620201576/5380742305932201]],
[82, [553/80, -294473/635180, -207130360353696711/2582103247360000, -3971389576/5179020201, -24743080/5179020201, -4657804375/5179020201]],
[83, [-12065/12396, -1581/1520, 6215830249009663/355017949286400, 812937165464036006213895/864745895259187110399737, 591519768111748983750888/864745895259187110399737, -57810716855047169409080/864745895259187110399737]],
[84, [-12065/12396, -1581/1520, -6215830249009663/355017949286400, 2164632/44310257, -31669120/44310257, -41084175/44310257]],
[85, [-12065/12396, 21792/5035, 33451262757025519/1947744960049800, 2164632/44310257, -31669120/44310257, 41084175/44310257]],
[86, [-12065/12396, 21792/5035, -33451262757025519/1947744960049800, -66491673395168374249746120/123188180833923372056627153, -118194421251475239056505903/123188180833923372056627153, 62831773759131557571594880/123188180833923372056627153]],
[87, [1744/495, 135/1208, 180347147759/178778080800, -435210480720/521084370137, 372623278887/521084370137, -369168502640/521084370137]],
[88, [1744/495, 135/1208, -180347147759/178778080800, -5819035124295/7082388012473, 4408757988560/7082388012473, -5611660306848/7082388012473]],
[89, [1744/495, -977657/480240, 162765645365418367/28255113936720000, -4408757988560/7082388012473, 5819035124295/7082388012473, 5611660306848/7082388012473]],
[90, [1744/495, -977657/480240, -162765645365418367/28255113936720000, -372623278887/521084370137, 435210480720/521084370137, 369168502640/521084370137]],
[91, [-3168/1553, -857/3696, 41906266886375/2353308160896, 19031674138785/27497822498977, 25762744660064/27497822498977, -2054845288320/27497822498977]],
[92, [-3168/1553, -857/3696, -41906266886375/2353308160896, 2927198165920/6382441853233, -613935345969/6382441853233, -6310500741600/6382441853233]],
[93, [-3168/1553, 980785/175296, 5796392699440292705/37055862675228672, 613935345969/6382441853233, -2927198165920/6382441853233, 6310500741600/6382441853233]],
[94, [-3168/1553, 980785/175296, -5796392699440292705/37055862675228672, -25762744660064/27497822498977, -19031674138785/27497822498977, 2054845288320/27497822498977]],
[95, [-1376/705, 14337/340, 22737959090039/1689885000, 148739531603136/230791363907489, 32467583677535/230791363907489, 220093974949320/230791363907489]],
[96, [-1376/705, 14337/340, -22737959090039/1689885000, -36295982895/39871595729, -29676864960/39871595729, -11262039896/39871595729]],
[97, [-1376/705, -81065/89412, 73070745575924711/1986734608705800, 29676864960/39871595729, 36295982895/39871595729, 11262039896/39871595729]],
[98, [-1376/705, -81065/89412, -73070745575924711/1986734608705800, -32467583677535/230791363907489, -148739531603136/230791363907489, -220093974949320/230791363907489]],
[99, [-1152/2345, 19005/3688, 1161796506978039/37397065344800, 443873167360/597385645737, -142485966505/597385645737, 544848079888/597385645737]],
[100, [-1152/2345, 19005/3688, -1161796506978039/37397065344800, -468405247415/1682315502153, -1657554153472/1682315502153, 801719896720/1682315502153]],
[101, [-1152/2345, -15461/13160, 4644845269335303/476175972020000, 1657554153472/1682315502153, 468405247415/1682315502153, -801719896720/1682315502153]],
[102, [-1152/2345, -15461/13160, -4644845269335303/476175972020000, 142485966505/597385645737, -443873167360/597385645737, -544848079888/597385645737]],
[103, [2265/184, -68256/135125, 106975611923719719/309084384500000, -78558599440/820234293081, 814295112544/820234293081, 337210257575/820234293081]],
[104, [2265/184, -68256/135125, -106975611923719719/309084384500000, -7745659501403353894384/12214291847502204701241, -2120589250533219579335/12214291847502204701241, -11684173258429439467360/12214291847502204701241]],
[105, [-1245/5012, 248521/62784, 208445266940805505/99019353702334464, 39110088360/76973733409, -49796687200/76973733409, 71826977313/76973733409]],
[106, [-1245/5012, 248521/62784, -208445266940805505/99019353702334464, 12209879806944320496330055/34497456764264994703368889, -26621272474250391413865480/34497456764264994703368889, 30730370351168229154149048/34497456764264994703368889]],
[107, [-1245/5012, -267904/221337, 2252317127132864545/615318775951504968, 435117527990435060232042280/504068891841730072306483681, -106958136069417067994530335/504068891841730072306483681, -411177854471028470696556192/504068891841730072306483681]],
[108, [-1245/5012, -267904/221337, -2252317127132864545/615318775951504968, 39110088360/76973733409, -49796687200/76973733409, -71826977313/76973733409]],
[109, [1873/200, -51416/9425, 4964755565640087/1776612500000, 487814048600/26969608212297, 8528631804200/26969608212297, 26901926181047/26969608212297]],
[110, [1873/200, -51416/9425, -4964755565640087/1776612500000, -103028409596553328/103117303193818953, 24975412054750025/103117303193818953, -4092004076331400/103117303193818953]],
[111, [-97/400, 78065/484, 2352959771684697/37480960000, 87375622888246360/87486470529871881, 16306696482461560/87486470529871881, 21794572772239369/87486470529871881]],
[112, [-97/400, 78065/484, -2352959771684697/37480960000, -1313903832425/6014017311081, -6010589044544/6014017311081, -66822832760/6014017311081]],
[113, [-1425/412, -9/20, 3894577617/67897600, 1390400/2813001, 2767624/2813001, 673865/2813001]],
[114, [-1425/412, -9/20, -3894577617/67897600, 140976551/15434547801, -5821981400/15434547801, -15355831360/15434547801]],
[115, [-1425/412, 5728/215, 47763328987689/3923208200, 5821981400/15434547801, -140976551/15434547801, 15355831360/15434547801]],
[116, [-1425/412, 5728/215, -47763328987689/3923208200, -2767624/2813001, -1390400/2813001, -673865/2813001]]
以上的值只依赖于下面24个u值得到:
[-9/20, -29/12, -93/80, -400/37, -136/133, 201/4, -5/8, 233/60, -56/165, -125/92, -361/540, -817/660, -865/592, 553/80, -12065/12396, 1744/495, -3168/1553, -1376/705, -1152/2345, 2265/184, -1245/5012, 1873/200, -97/400, 553/80]
要想得到较小的有理解(20位以内),需要找到新的u值,观察上面已知的u值(都不是太大,且绝大数分母含因子4;负数较多;分子分母的质因子都是2(指数不超过7)、3(指数不超过5)、5,7,13,17,19,23,29,31,37,41...(指数不超过1);分子分母不同质因子总数在4~7;)
按以上的规则应该可以找到新的u值? |
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