周长、面积相等的本原三角形

wayne

感觉这种是有成效的。就是针对单个的椭圆曲线就有大量的手工处理的过程，没法大量处理，效率很低。

northwolves

lsr314 发表于 2019-11-26 15:24
有五个的，周长是$7546$，面积是$2522520$：
${1901,2772,2873},{1914,2723,2909},{1925,2693,2928},{201 ...

有6个的，周长是$20026$，面积是$8410920：{2108,8493,9425},{2173,8398,9455},{2261,8277,9488},{2418,8075,9533},{4123,6205,9698},{4588,5729,9709}$

northwolves

周长在1-34500，暂搜到16组周长面积都是整数的五个解的组合：

{{2842,267960,{{464,1157,1221},{541,1025,1276},{596,957,1289},{701,841,1300},{725,816,1301}}},
{3542,318780,{{421,1518,1603},{511,1375,1656},{583,1288,1671},{759,1096,1687},{851,1001,1690}}},
{5642,406224,{{310,2639,2693},{325,2604,2713},{868,1989,2785},{1085,1768,2789},{1183,1669,2790}}},
{5750,1062600,{{915,2323,2512},{1060,2091,2599},{1104,2035,2611},{1335,1771,2644},{1489,1610,2651}}},
{6314,757680,{{533,2849,2932},{957,2296,3061},{1237,2002,3075},{1312,1925,3077},{1517,1717,3080}}},
{7238,1302840,{{819,3196,3223},{1034,2809,3395},{1351,2444,3443},{1363,2431,3444},{1645,2134,3459}}},
{7546,2522520,{{1901,2772,2873},{1914,2723,2909},{1925,2693,2928},{2018,2525,3003},{2213,2288,3045}}},
{9790,1233540,{{534,4625,4631},{583,4490,4717},{890,4103,4797},{1157,3815,4818},{1925,3026,4839}}},
{15470,8632260,{{2821,6154,6495},{2975,5813,6682},{3145,5551,6774},{3395,5224,6851},{4046,4495,6929}}},
{15778,9669660,{{3170,6279,6329},{3289,5887,6602},{3833,5106,6839},{4094,4809,6875},{4301,4589,6888}}},
{17710,124 32420,{{3850,6463,7397},{3887,6398,7425},{4025,6182,7503},{4760,5291,7659},{4968,5075,7667}}},
{21658,12994800,{{2989,8840,9829},{3329,8333,9996},{3332,8329,9997},{4704,6749,10205},{5304,6125,10229}}},
21970,14479920,{{3145,9217,9608},{3488,8537,9945},{4049,7800,10121},{4797,6953,10220},{4916,6825,10229}}},
{21970,14054040,{{3047,9225,9698},{3120,9041,9809},{3425,8528,10017},{3978,7817,10175},{4937,6750,10283}}},
{22610,19399380,{{4389,9061,9160},{4408,8925,9277},{4675,8302,9633},{5585,7106,9919},{5950,6707,9953}}},
{26962,22648080,{{4148,10933,11881},{4641,10121,12200},{5551,9001,12410},{5837,8681,12444},{6953,7503,12506}}}};

wayne

http://www.maroon.dti.ne.jp/fermat/dioph312e.html
这个链接给出了 周长和面积都相等的海伦三角形,有50组解的情况,但不是本原三角形.三边的公约数不为1, 化成本原三角形后,也只是三组解.
{{174,91,125},390,5460},
{{175,104,111},390,5460},
{{146,75,169},390,5460}

wayne

用椭圆曲线搜索, 进度好像有点慢,.

1. {133042,131711580}->{{133042,131711580,8925,58136,65981},{133042,131711580,4151,64090,64801},{133042,131711580,5031,62561,65450},{133042,131711580,11441,55510,66091},{133042,131711580,12971,53937,66134}}
   
2. {26962,22648080}->{{26962,22648080,5837,8681,12444},{26962,22648080,4148,10933,11881},{26962,22648080,5551,9001,12410},{26962,22648080,6953,7503,12506},{26962,22648080,4641,10121,12200}}
   
3. {61180,60843510}->{{61180,60843510,4580,27531,29069},{61180,60843510,4427,27853,28900},{61180,60843510,4291,28405,28484},{61180,60843510,5175,26612,29393}}
   
4. {36890,63303240}->{{36890,63303240,11165,11713,14012},{36890,63303240,11284,11581,14025},{36890,63303240,11009,11900,13981},{36890,63303240,10540,12637,13713}}
   
5. {32890,24864840}->{{32890,24864840,4538,12765,15587},{32890,24864840,4125,13286,15479},{32890,24864840,4025,13421,15444},{32890,24864840,3404,14625,14861}}
   
6. {21114,5123664}->{{21114,5123664,4733,5916,10465},{21114,5123664,5083,5565,10466},{21114,5123664,3485,7176,10453},{21114,5123664,1093,9821,10200}}
   
7. {2431702,27283696440}->{{2431702,27283696440,375627,842174,1213901},{2431702,27283696440,85436,1137311,1208955},{2431702,27283696440,94820,1127191,1209691}}
   
8. {1882444,106042776630}->{{1882444,106042776630,379525,630052,872867},{1882444,106042776630,309225,715892,857327},{1882444,106042776630,266305,802807,813332}}
   
9. {1395394,65857015224}->{{1395394,65857015224,252945,526301,616148},{1395394,65857015224,264860,508913,621621},{1395394,65857015224,231855,572033,591506}}
   
10. {1388618,28269485244}->{{1388618,28269485244,297561,406840,684217},{1388618,28269485244,119371,595122,674125},{1388618,28269485244,105435,612877,670306}}
    
11. {352002,5885473440}->{{352002,5885473440,106689,120176,125137},{352002,5885473440,109121,115600,127281},{352002,5885473440,109330,115311,127361}}
    
12. {290444,3727122630}->{{290444,3727122630,72772,106931,110741},{290444,3727122630,84747,88228,117469},{290444,3727122630,79827,93892,116725}}
    
13. {212058,1853386920}->{{212058,1853386920,53865,68884,89309},{212058,1853386920,56933,65296,89829},{212058,1853386920,52164,71069,88825}}
    
14. {196042,1834953120}->{{196042,1834953120,60581,67536,67925},{196042,1834953120,62700,63461,69881},{196042,1834953120,62531,63650,69861}}
    
15. {153140,73966620}->{{153140,73966620,6213,70525,76402},{153140,73966620,11687,64978,76475},{153140,73966620,35929,40690,76521}}
    
16. {112746,451434984}->{{112746,451434984,22724,40357,49665},{112746,451434984,20425,44229,48092},{112746,451434984,19952,45885,46909}}
    
17. {104650,467157600}->{{104650,467157600,24518,39525,40607},{104650,467157600,29757,31400,43493},{104650,467157600,27125,34469,43056}}
    
18. {71610,185040240}->{{71610,185040240,15950,23901,31759},{71610,185040240,12957,28985,29668},{71610,185040240,14725,25469,31416}}
    
19. {28842,11306064}->{{28842,11306064,6693,7904,14245},{28842,11306064,3445,11229,14168},{28842,11306064,2024,12853,13965}}
    
20. {23738,24212760}->{{23738,24212760,6225,7789,9724},{23738,24212760,5629,8979,9130},{23738,24212760,5644,8877,9217}}
    
21. {23370,14395920}->{{23370,14395920,3325,9221,10824},{23370,14395920,4645,7667,11058},{23370,14395920,3116,9529,10725}}
    
22. {17738,9365664}->{{17738,9365664,2597,7240,7901},{17738,9365664,2940,6697,8101},{17738,9365664,2896,6757,8085}}
    
23. {14098,9304680}->{{14098,9304680,4134,4739,5225},{14098,9304680,4389,4409,5300},{14098,9304680,4081,4849,5168}}
    
24. {12194,438984}->{{12194,438984,871,5233,6090},{12194,438984,481,5628,6085},{12194,438984,1876,4225,6093}}
    
25. {9130,3067680}->{{9130,3067680,1925,3221,3984},{9130,3067680,2075,3029,4026},{9130,3067680,2255,2822,4053}}
    
26. {6954,389424}->{{6954,389424,285,3253,3416},{6954,389424,488,3021,3445},{6954,389424,915,2581,3458}}
    

复制代码

mathe

椭圆曲线求解关键问题还在于解的顺序不容易保证。

mathe

35594 26666640 :5 {637 5280 11880}{672 4840 12285}{715 4410 12672}{847 3510 13440}{1120 2520 14157}
35728 33093060 :5 {855 7714 9295}{858 7581 9425}{1050 4901 11913}{1218 3971 12675}{1885 2394 13585}
36366 10182480 :5 {96 4312 13775}{152 2400 15631}{245 1408 16530}{319 1064 16800}{550 609 17024}
37210 15679440 :5 {170 6528 11907}{245 3672 14688}{272 3213 15120}{324 2601 15680}{405 2016 16184}
38522 48537720 :5 {1568 8343 9350}{1617 7344 10300}{1785 5940 11536}{1836 5665 11760}{2475 3808 12978}
38642 13310640 :5 {112 6384 12825}{133 4788 14400}{152 3969 15200}{196 2880 16245}{432 1200 17689}
40326 53633580 :5 {1880 6370 11913}{1900 6251 12012}{2068 5460 12635}{2184 5054 12925}{2310 4693 13160}
48070 102965940 :5 {4760 9177 10098}{4968 8085 10982}{5060 7803 11172}{5202 7448 11385}{5814 6440 11781}
48230 26912340 :5 {217 9858 14040}{265 6552 17298}{424 3510 20181}{728 1922 21465}{1113 1240 21762}
49818 104984880 :5 {4186 9075 11648}{4224 8855 11830}{4400 8112 12397}{4485 7840 12584}{4928 6825 13156}
52250 68468400 :5 {1152 12103 12870}{1330 8064 16731}{1672 5733 18720}{2080 4389 19656}{2457 3648 20020}
66062 124857180 :5 {1960 14790 16281}{1972 14175 16884}{3402 5829 23800}{3417 5800 23814}{4050 4824 24157}
67298 52492440 :5 {299 14630 18720}{300 14421 18928}{352 9975 23322}{624 4620 28405}{1560 1729 30360}
68034 104772360 :5 {1218 14399 18400}{1265 12760 19992}{1400 10353 22264}{3025 3944 27048}{3360 3542 27115}
69046 159496260 :5 {3036 13475 18012}{3192 11850 19481}{3630 9480 21413}{3933 8470 22120}{5225 6072 23226}
70642 232792560 :6 {8624 12825 13872}{8640 12716 13965}{8721 12320 14280}{9248 11025 15048}{9537 10584 15200}{9639 10450 15232}
71162 93933840 :5 {825 16192 18564}{920 11781 22880}{1173 8008 26400}{1518 5775 28288}{2240 3740 29601}
73034 108228120 :5 {1020 17017 18480}{1089 13328 22100}{1225 10608 24684}{1760 6435 28322}{1848 6069 28600}
74074 26666640 :5 {64 12025 24948}{77 8880 28080}{126 4736 32175}{312 1760 34965}{572 945 35520}
我后面三个数列的是{x,y,z}而不是边长

mathe

77714 55954080 :6 {216 18816 19825}{256 11700 26901}{360 7137 31360}{637 3660 34560}{768 2989 35100}{832 2745 35280}
79534 71580600 :5 {405 11362 28000}{437 10080 29250}{672 5750 33345}{1120 3250 35397}{1512 2375 35880}
79534 262462200 :5 {6160 15732 17875}{6578 13125 20064}{6875 12144 20748}{7392 10925 21450}{8625 9120 22022}
84968 147207060 :5 {1197 20475 20812}{1485 11739 29260}{1505 11495 29484}{1548 11011 29925}{1881 8428 32175}
85690 187146960 :6 {2280 13013 27552}{2337 12480 28028}{2688 10127 30030}{2704 10045 30096}{3211 8064 31570}{3234 7995 31616}
85974 189142800 :5 {2047 16940 24000}{2112 15575 25300}{3680 7000 32307}{3750 6853 32384}{4235 6000 32752}
90610 167447280 :5 {1312 18513 25480}{1344 17220 26741}{1870 9856 33579}{2009 8976 34320}{3872 4305 37128}
91698 282429840 :7 {3968 20706 21175}{4092 17600 24157}{4480 14399 26970}{4675 13398 27776}{5104 11760 28985}{6545 8680 30624}{7161 7888 30800}
96746 167062896 :5 {1053 20216 27104}{1216 14553 32604}{2184 6688 39501}{3024 4693 40656}{3249 4356 40768}
97498 116854920 :5 {650 11907 36192}{1080 6264 41405}{1260 5265 42224}{1352 4872 42525}{2430 2639 43680}
106106 324684360 :7 {3213 23320 26520}{3240 22253 27560}{3600 17017 32436}{4081 13872 35100}{4620 11713 36720}{4624 11700 36729}{5088 10395 37570}

发现7组的了

mathe

124930 379819440 :6 {2601 27104 32760}{2772 22253 37440}{3024 18785 40656}{3978 12672 45815}{4165 11968 46332}{6048 7865 48552}
133042 131711580 :5 {387 12584 53550}{430 11011 55080}{540 8385 57596}{1071 3960 61490}{1720 2431 62370}
140182 277560360 :5 {935 30096 39060}{1178 18513 50400}{1680 11495 56916}{2380 7695 60016}{2907 6200 60984}
140182 555120720 :5 {4032 32300 33759}{4760 20691 44640}{5016 19040 46035}{7425 11594 51072}{8064 10602 51425}
141722 701523900 :5 {6996 26740 37125}{7500 22869 40492}{7875 21010 41976}{10123 15000 45738}{11880 12606 46375}
150898 170117640 :6 {288 28611 46550}{297 26600 48552}{374 17955 57120}{425 15048 59976}{1445 3780 70224}{1710 3179 70560}
151810 439945380 :5 {1862 35955 38088}{1880 33327 40698}{2793 15980 57132}{3243 13230 59432}{3570 11799 60536}
152626 164180016 :6 {273 25704 50336}{468 11781 64064}{561 9504 66248}{884 5733 69696}{1573 3136 71604}{1859 2646 71808}
154882 613332720 :6 {3848 27225 46368}{3960 25641 47840}{4477 20800 52164}{4680 19481 53280}{5106 17280 55055}{8064 10175 59202}
157250 174358800 :6 {252 38080 40293}{259 32670 45696}{484 12096 66045}{833 6512 71280}{1056 5049 72520}{2205 2368 74052}
162266 1167206040 :5 {15840 30258 35035}{16810 26208 38115}{17220 25168 38745}{18081 23452 39600}{19305 21648 40180}
183034 1102978800 :5 {8000 32697 50820}{8477 29040 54000}{8712 27680 55125}{10584 20933 60000}{14112 15125 62280}
193154 324498720 :5 {476 43776 52325}{480 41952 54145}{741 19200 76636}{1225 10488 84864}{3458 3519 89600}
193154 892371480 :5 {3872 41990 50715}{3920 39780 52877}{3927 39520 53130}{4485 29260 62832}{5152 23595 67830}
197846 1606268664 :5 {15561 33858 49504}{17556 27664 53703}{17689 27378 53856}{20111 23408 55404}{21216 22113 55594}

northwolves

周长p,面积s都是整数的n个解的最小组合:

n=1   12                6        {{3,4,5}}}

n=2   70                210        {{20,21,29},{17,25,28}}

n=3   210                840        {{21,89,100},{25,84,101},{35,73,102}}

n=4   2002        120120        {{286,841,875},{308,793,901},{377,701,924},{521,546,935}}

n=5   2842        267960        {{464,1157,1221},{541,1025,1276},{596,957,1289},{701,841,1300},{725,816,1301}}

n=6   20026        8410920        {{2108,8493,9425},{2173,8398,9455},{2261,8277,9488},{2418,8075,9533},{4123,6205,9698},{4588,5729,9709}}

n=7   71456     132372240    {{8558,30940,31958},{8653,30624,32179},{10378,27786,33292},{10933,27083,33440},{11902,25926,33628},{16878,20566,34012},{17138,20300,34018}}

n=8