周长、面积相等的本原三角形

mathe · 发表于 2025-7-12 05:33:47

213378 388347960 :5 {2562 5600 98527}{689 25480 80520}{583 33306 72800}{1859 7840 96990}{528 41236 64925}
213774 1945343400 :5 {22120 28392 56375}{18960 35875 52052}{20787 30800 55300}{18942 35945 52000}{17875 44240 44772}
219674 1502570160 :5 {8449 38988 62400}{9585 30940 69312}{8512 38340 62985}{12996 20800 76041}{8320 40470 61047}
222404 1367656290 :5 {6251 44550 60401}{12502 16335 82365}{7315 31977 71910}{6137 48195 56870}{8925 24111 78166}
225998 921274200 :5 {8349 9450 95200}{2499 48400 62100}{7344 10780 94875}{2464 52785 57750}{2800 36300 73899}
231322 870233364 :5 {2706 28728 84227}{4712 14391 96558}{2046 51623 61992}{3458 20691 91512}{6776 9747 99138}
235586 2009367360 :5 {20295 22528 74970}{16065 29568 72160}{12705 44608 60480}{14520 34153 69120}{14336 34850 68607}
243602 1551109560 :5 {6253 44400 71148}{6105 47320 68376}{7007 35594 79200}{8450 27104 86247}{5880 57720 58201}
249458 1257268320 :5 {7308 17325 100096}{5152 26400 93177}{5950 22011 96768}{3465 56304 64960}{4312 34017 86400}
257114 1619818200 :5 {9889 21168 97500}{11600 17732 99225}{9625 21840 97092}{7056 32500 89001}{6750 34720 87087}
257278 2207445240 :5 {14382 32032 82225}{12784 38115 77740}{14100 32912 81627}{18564 23595 86480}{11830 43945 72864}
263074 2951690280 :6 {27692 34485 69360}{22287 51170 58080}{24225 41624 65688}{22990 46240 62307}{22440 49665 59432}{22253 51600 57684}
276138 961440480 :5 {1508 53361 83200}{4901 11200 121968}{1872 35525 100672}{1792 38025 98252}{3584 15730 118755}
282854 1211746536 :5 {2967 33264 105196}{4816 18207 118404}{3289 28896 109242}{2576 41327 97524}{6664 12771 121992}

wayne · 发表于 2025-7-12 09:10:00

我给一下 mathe没有的。按照周长排序。

Mathematica太慢了。用C++/ flint实现了一遍。遗憾的是flint不支持 GMP的 mpz_class，导致很多公式都需要用C语言接口的方式实现，太boring了

wayne · 发表于 2025-7-13 01:11:00

多线程搞起来了，然后8组解蹦出来了
{8,{4/5,132/221},{721786,13338605280},{{188105,189428,344253},{179133,198458,344195},{124338,256523,340925},{116093,266029,339664},{91448,299013,331325},{88253,304980,328553},{85981,310493,325312},{85618,311627,324541}}}

wayne · 发表于 2025-7-14 11:49:44

跑了一晚上,进度并不乐观。我给出当前所有的6,7,8的解
{{3, 15865}, {4, 1457}, {5, 186}, {6, 32}, {7, 7}, {8, 1}}

{8, {4/5, 132/221}, {721786, 13338605280}, {{188105, 189428, 344253}, {179133, 198458, 344195}, {124338, 256523, 340925}, {116093, 266029, 339664}, {91448, 299013, 331325}, {88253, 304980, 328553}, {85981, 310493, 325312}, {85618, 311627, 324541}}}
{7, {5/17, 77/377}, {12806690, 2414009838240}, {{2146001, 4362064, 6298625}, {2017600, 4494449, 6294641}, {1956672, 4557553, 6292465}, {1620369, 4910425, 6275896}, {1499281, 5040200, 6267209}, {1224465, 5344903, 6237322}, {819151, 5924625, 6062914}}}
{7, {17/40, 52/155}, {1866634, 53871057240}, {{313973, 636244, 916417}, {289850, 661227, 915557}, {196581, 761267, 908786}, {189210, 769637, 907787}, {165308, 797909, 903417}, {150249, 817292, 899093}, {132452, 844657, 889525}}}
{7, {19/55, 44/155}, {1002478, 11578620900}, {{180079, 325910, 496489}, {163286, 342953, 496239}, {122322, 384989, 495167}, {112739, 394979, 494760}, {89355, 419864, 493259}, {73625, 437414, 491439}, {61864, 451651, 488963}}}
{7, {29/36, 21/44}, {466378, 7639271640}, {{109650, 147059, 209669}, {105425, 151844, 209109}, {97053, 161936, 207389}, {95030, 164549, 206799}, {92191, 168389, 205798}, {84194, 181475, 200709}, {81141, 192473, 192764}}}
{7, {37/99, 25/111}, {440818, 4010252400}, {{94184, 132825, 213809}, {93249, 133784, 213785}, {87584, 139625, 213609}, {64349, 164409, 212060}, {53792, 176617, 210409}, {40940, 196409, 203469}, {40409, 198513, 201896}}}
{7, {3/13, 21/40}, {106106, 324684360}, {{26533, 29733, 49840}, {25493, 30800, 49813}, {20617, 36036, 49453}, {17953, 39181, 48972}, {16333, 41340, 48433}, {16324, 41353, 48429}, {15483, 42658, 47965}}}
{7, {16/17, 22/31}, {91698, 282429840}, {{24674, 25143, 41881}, {21692, 28249, 41757}, {18879, 31450, 41369}, {18073, 32451, 41174}, {16864, 34089, 40745}, {15225, 37169, 39304}, {15049, 37961, 38688}}}
{6, {28/37, 143/152}, {25470874, 14024772641880}, {{6218157, 6925142, 12327575}, {3822316, 9438737, 12209821}, {3357417, 9974525, 12138932}, {3154649, 10222212, 12094013}, {3107852, 10281245, 12081777}, {2491245, 11280227, 11699402}}}
{6, {12/59, 93/95}, {24585418, 20305834288740}, {{6251000, 7113689, 11220729}, {5724770, 7673549, 11187099}, {5351654, 8091239, 11142525}, {4385194, 9348019, 10852205}, {4171064, 9738729, 10675625}, {4160975, 9761054, 10663389}}}
{6, {12/37, 164/315}, {13777394, 4964821701840}, {{3429241, 3793647, 6554506}, {2597452, 4666477, 6513465}, {2486585, 4789292, 6501517}, {2471193, 4806529, 6499672}, {1874457, 5545537, 6357400}, {1714297, 5823763, 6239334}}}
{6, {13/24, 95/268}, {13355914, 4582948329960}, {{2923557, 4076414, 6355943}, {2379677, 4661085, 6315152}, {2307767, 4741925, 6306222}, {2042557, 5052944, 6260413}, {1777061, 5405071, 6173782}, {1647932, 5622393, 6085589}}}
{6, {37/48, 23/62}, {8779138, 2883420084720}, {{2277065, 2618849, 3883224}, {2176118, 2728651, 3874369}, {2090489, 2825960, 3862689}, {1884225, 3084734, 3810179}, {1769784, 3259625, 3749729}, {1723409, 3348625, 3707104}}}
{6, {9/38, 140/153}, {4326262, 689562900180}, {{1145187, 1255994, 1925081}, {1039121, 1370850, 1916291}, {947903, 1480031, 1898328}, {936675, 1494521, 1895066}, {919581, 1517222, 1889459}, {817171, 1694341, 1814750}}}
{6, {29/50, 45/98}, {3526922, 234893005200}, {{863736, 942061, 1721125}, {676549, 1132537, 1717836}, {572320, 1241461, 1713141}, {394461, 1439560, 1692901}, {367336, 1473381, 1686205}, {334225, 1518661, 1674036}}}
{6, {58/77, 22/83}, {3355358, 403045602960}, {{842537, 1008367, 1504454}, {773629, 1084658, 1497071}, {771319, 1087320, 1496719}, {761192, 1099087, 1495079}, {711399, 1159759, 1484200}, {687599, 1191050, 1476709}}}
{6, {57/88, 95/184}, {2981766, 174492946320}, {{698763, 831203, 1451800}, {394051, 1151733, 1435982}, {305083, 1260584, 1416099}, {302498, 1264195, 1415073}, {272527, 1312131, 1397108}, {265098, 1328195, 1388473}}}
{6, {49/76, 115/124}, {1778590, 97260415560}, {{455791, 495874, 826925}, {435083, 517140, 826367}, {422807, 529943, 825840}, {371316, 585599, 821675}, {361075, 597151, 820364}, {271062, 719663, 787865}}}
{6, {13/63, 5/119}, {1671202, 23162859720}, {{338181, 501281, 831740}, {175865, 665456, 829881}, {172533, 668876, 829793}, {124176, 719177, 827849}, {94525, 751361, 825316}, {57551, 805154, 808497}}}
{6, {5/13, 27/403}, {953498, 12634581960}, {{241269, 241525, 470704}, {236223, 246574, 470701}, {195517, 287524, 470457}, {128557, 356167, 468774}, {81189, 407845, 464464}, {56637, 446212, 450649}}}
{6, {13/44, 112/299}, {892078, 29224475280}, {{241703, 252839, 397536}, {224279, 271124, 396675}, {203524, 294679, 393875}, {188600, 313599, 389879}, {181047, 324392, 386639}, {177719, 329615, 384744}}}
{6, {11/26, 85/156}, {848470, 13831757940}, {{166120, 269399, 412951}, {155651, 280245, 412574}, {106346, 333529, 408595}, {94829, 347171, 406470}, {72488, 382375, 393607}, {71519, 388056, 388895}}}
{6, {39/41, 65/111}, {816146, 26736942960}, {{231609, 233248, 351289}, {204709, 262564, 348873}, {199060, 269493, 347593}, {184265, 290228, 341653}, {180768, 296185, 339193}, {174973, 308812, 332361}}}
{6, {19/24, 21/136}, {788766, 17226649440}, {{186175, 230223, 372368}, {169423, 247775, 371568}, {124113, 299663, 364990}, {111503, 317460, 359803}, {104975, 329448, 354343}, {102000, 338143, 348623}}}
{6, {6/23, 35/78}, {761254, 21101960880}, {{194166, 227411, 339677}, {169845, 254942, 336467}, {168107, 257075, 336072}, {141542, 299117, 320595}, {139859, 305118, 316277}, {139232, 310947, 311075}}}
{6, {24/41, 33/52}, {659362, 3600116520}, {{119884, 211381, 328097}, {53669, 278812, 326881}, {52521, 280016, 326825}, {25585, 311026, 322751}, {24467, 312881, 322014}, {23375, 315121, 320866}}}
{6, {7/13, 127/133}, {617474, 14782327560}, {{171196, 176437, 269841}, {162052, 185997, 269425}, {136017, 217765, 263692}, {134640, 219817, 263017}, {123937, 243586, 249951}, {123825, 244477, 249172}}}
{6, {7/23, 9/41}, {465842, 6456570120}, {{123196, 124813, 217833}, {119301, 128740, 217801}, {106641, 141841, 217360}, {69121, 188272, 208449}, {66871, 193154, 205817}, {65481, 198436, 201925}}}
{6, {15/28, 39/68}, {452998, 1386173880}, {{95004, 132179, 225815}, {88556, 138643, 225799}, {28823, 199251, 224924}, {17654, 211565, 223779}, {14934, 215075, 222989}, {12707, 219317, 220974}}}
{6, {9/46, 47/168}, {451858, 2846705400}, {{90804, 138149, 222905}, {67925, 161529, 222404}, {45929, 184851, 221078}, {42389, 188804, 220665}, {37145, 194909, 219804}, {33649, 199280, 218929}}}
{6, {8/9, 80/121}, {418418, 4669544880}, {{104377, 115434, 198607}, {81396, 139513, 197509}, {75289, 146289, 196840}, {64372, 159237, 194809}, {59293, 166009, 193116}, {53599, 175450, 189369}}}
{6, {11/28, 5/34}, {376618, 3084501420}, {{76925, 117819, 181874}, {58088, 137921, 180609}, {47025, 150824, 178769}, {43814, 154989, 177815}, {38159, 163982, 174477}, {36729, 167960, 171929}}}
{6, {18/47, 57/112}, {330410, 2164846320}, {{66223, 103637, 160550}, {42901, 128649, 158860}, {40885, 130989, 158536}, {36613, 136197, 157600}, {33892, 139825, 156693}, {31960, 142709, 155741}}}
{6, {11/34, 136/209}, {263074, 2951690280}, {{73853, 79937, 109284}, {73457, 80367, 109250}, {72105, 81872, 109097}, {69230, 85297, 108547}, {65849, 89913, 107312}, {62177, 97052, 103845}}}
{6, {10/21, 112/255}, {157250, 174358800}, {{38332, 40545, 78373}, {32929, 45955, 78366}, {12580, 66529, 78141}, {7345, 72113, 77792}, {6105, 73576, 77569}, {4573, 76257, 76420}}}
{6, {13/22, 11/74}, {154882, 613332720}, {{31073, 50216, 73593}, {29601, 51800, 73481}, {25277, 56641, 72964}, {24161, 57960, 72761}, {22386, 60161, 72335}, {18239, 67266, 69377}}}
{6, {8/13, 8/55}, {85690, 187146960}, {{15293, 29832, 40565}, {14817, 30365, 40508}, {12815, 32718, 40157}, {12749, 32800, 40141}, {11275, 34781, 39634}, {11229, 34850, 39611}}}
{6, {26/45, 2/49}, {77714, 55954080}, {{19032, 20041, 38641}, {11956, 27157, 38601}, {7497, 31720, 38497}, {4297, 35197, 38220}, {3757, 35868, 38089}, {3577, 36112, 38025}}}
{6, {45/104, 80/117}, {70642, 232792560}, {{21449, 22496, 26697}, {21356, 22605, 26681}, {21041, 23001, 26600}, {20273, 24296, 26073}, {20121, 24737, 25784}, {20089, 24871, 25682}}}
{6, {5/8, 15/31}, {20026, 8410920}, {{4588, 5729, 9709}, {4123, 6205, 9698}, {2418, 8075, 9533}, {2261, 8277, 9488}, {2173, 8398, 9455}, {2108, 8493, 9425}}}

复制代码

简述一下我的思路,设三角形三边分别是$[a,b,c]=[\frac{2 R u}{u^2+1},\frac{2 R v}{v^2+1},\frac{2 R w}{w^2+1}]$,那么w可以用u,v表达,为了遏制对称性带来的多个解. 不妨设$0<u<v<1, v<w$,那么,根据sin(A+B)的展开,w有两个解.$w_1=\frac{u v+1}{v-u}, w_2=\frac{1-u v}{u+v}$,
对于解 1) 三边表达式是$[a,b,c]=[d u (1 + v^2),d (1 + u^2) v,d (v - u) (1 + u v)]$,周长是$2 d v (u v+1)$,面积是 $d^2 u v (v-u) (1+u v)$,外接圆直径是$2R=d (u^2+1) (v^2+1)$
对于解 2) 三边表达式是$[a,b,c]=[d u (1 + v^2),d (1 + u^2) v,d (u + v) (1 - u v)]$,周长是$2 d (u+v)$,面积是 $d^2 u v (u + v) (1 - u v)$,外接圆直径是$2R=d (u^2+1) (v^2+1)$

其实,解1和解2是关于v的倒数变换.

然后我们利用根据海伦公式得知$k=\frac{16S^2}{p} =(a+b-c)(a-b+c)(-a+b+c)$,就是分解$k_1=\frac{16S_1^2}{p_1} = 8 d^3 u^2 v (u-v)^2 (u v+1)$ , $k_2=\frac{16S_2^2}{p_2} = 8 d^3 u^2 v^2 (u+v) (1-u v)^2$

分析到这一步, 我发现我的代码还有很大的优化力度,原则上普通PC,一天之内计算出 9组,10组,11组都是值得期待的.
比如,u,v互质.并且存储k值,用来去重,避免重复计算.
...

继续设$u\to \frac{n}{m},v\to \frac{p}{q},d\to (m q)^2$ ,可以得到两个解分别是
$[k,a,b,c,p,S,w]=[8 m n^2 p q^2 (m q+n p) (m p-n q)^2,m n \left(p^2+q^2\right),p q \left(m^2+n^2\right),(m q+n p) (m p-n q),2 m p (m q+n p),m n p q (m q+n p) (m p-n q),\frac{m q+n p}{m p-n q}]$
$[k,a,b,c,p,S,w]=[8 m n^2 p^2 q (m q-n p)^2 (m p+n q),m n \left(p^2+q^2\right),p q \left(m^2+n^2\right),(m q-n p) (m p+n q),2 m q (m p+n q),m n p q (m q-n p) (m p+n q),\frac{m q-n p}{m p+n q}]$
而这两个表达式在形式上其实是$p,q$的置换. 所以我们只需要让$v=\frac{p}{q}$可以大于1,也可以小于1就行, 也就是约束条件成了$0<u<1 \and u<v$

mathe · 发表于 2025-7-14 15:56:37

上面表达式如果我们设
$gcd(n,p)=u_{00}, gcd(m,q)=u_{11}, gcd(n,q)=u_{01}, gcd(m,p)=u_{10}$
那么可以得到$n=u_{00}u_{01}n_0, m=u_{10}u_{01}m_0,p=u_{00}u_{10}p_0, q=u_{01}u_{11}q_0$
于是可以消去三边公因子$u_{11}u_{10}u_{01}u_{00}$,余下
$a=m_0n_0(p^2+q^2), b=p_0q_0(m^2+n^2), c=(u_{11}^2m_0q_0-u_{00}^2p_0n_0)(u_{10}^2p_0m_0+u_{01}^2q_0n_0)$
当然我们无法继续判断gcd(a,b,c)这时是否为1，但是可以知道如果公因子大于1，必然是$gcd(p^2+q^2,m^2+n^2)$的因子，所以必然所有素因子模4余1或是2.
另外可以计算得到半周长$L=u_{11}^2q_0m_0(u_{10}^2p_0m_0+u_{01}^2q_0n_0)$，内切圆半径长度$\frac{S}{L}=u_{01}^2q_0n_0(u_{11}^2q_0m_0-u_{00}^2p_0n_0)$ (不确定是否完全正确，比较奇怪L和$u_{00}$无关）。
当然如果上面gcd(a,b,c)>1, 那么内切圆半径还需要初一这个公约数，就不一定是整数了，这是唯一比较麻烦的地方。
不然我们就应该可以试一试穷举L的上面因子分解形式。

mathe · 发表于 2025-7-14 16:56:22

前面顺序搜索代码应该是手搓开根号代码有bug,现在改成用gmp的，可以继续跑下去了
309442 4367923560 :5 {35360 51051 68310}{40392 42504 71825}{34320 54621 65780}{39325 43792 71604}{34848 52598 67275}
312550 2486647800 :5 {7191 67184 81900}{10982 31725 113568}{7650 55601 93024}{9126 40749 106400}{8400 46436 101439}
316250 3522519000 :5 {23322 33075 101728}{17017 51408 89700}{18700 43953 95472}{21675 36064 100386}{15444 68425 74256}
318274 2358410340 :5 {16606 16731 125800}{11385 25012 122740}{7752 40755 110630}{8619 35150 115368}{7436 43401 108300}
318734 2934804600 :5 {21525 21600 116242}{9975 61992 87400}{13300 37392 108675}{9747 67620 82000}{12250 41952 105165}
319010 1768591440 :5 {11753 12320 135432}{7227 20608 131670}{7980 18469 133056}{3591 53130 102784}{5544 28105 125856}
319906 3313995300 :5 {14210 51000 94743}{15660 43500 100793}{15138 45815 99000}{13068 61625 85260}{18375 35090 106488}
325486 2792669880 :5 {15268 25795 121680}{13880 28743 120120}{8675 56784 97284}{8400 61347 92996}{16800 23249 122694}
330410 2164846320 :6 {6345 36556 122304}{6669 34216 124320}{8512 25380 131313}{4655 61568 98982}{9464 22496 133245}{7605 29008 128592}
334970 2194723440 :5 {4816 55965 106704}{4446 68224 94815}{7488 29412 130585}{4563 63210 99712}{5040 51376 111069}
348194 2909907000 :5 {8550 48672 116875}{7125 71500 95472}{15147 23750 135200}{14872 24225 135000}{8112 53125 112860}
352594 2342506320 :5 {8225 26784 141288}{5922 40455 129920}{8990 24192 143115}{11774 18048 146475}{4205 84672 87420}
376618 3084501420 :6 {10494 33320 144495}{9540 37485 141284}{13832 24327 150150}{6435 70490 111384}{16380 20349 151580}{7700 50388 130221}
389266 6227200980 :5 {41800 43659 109174}{32186 61047 101400}{37905 48620 108108}{39270 46683 108680}{29988 70785 93860}
401882 4284680400 :7 {11070 67375 122496}{17226 35875 147840}{17820 34496 148625}{22000 27405 151536}{13440 49126 138375}{10500 76096 114345}{12000 58261 130680}
412282 4625804040 :5 {15857 45084 145200}{20724 32825 152592}{14025 53328 138788}{15028 48400 142713}{11440 77265 117436}
412918 6434468040 :5 {28336 59823 118300}{34385 46410 125664}{24276 85008 97175}{26208 68425 111826}{39039 40460 126960}

northwolves · 发表于 2025-7-14 22:32:45

周长为34322，面积为整数的本原三角形有779个，如何计算有1000个本原三角形的最小周长？10000个呢？

mathe · 发表于 2025-7-15 10:41:24

验证了
721786 13338605280 :8 {21229 94864 244800}{36352 49266 275275}{35581 50400 274912}{16698 162435 181760}{16640 171465 172788}{19968 104370 236555}{29568 61880 269445}{32340 55913 272640}
是最小的8组结果

mathe · 发表于 2025-7-15 11:15:42

分布很不均匀，40多万处好结果比较多
416806 1444232790 :5 {943 91630 115830}{5474 9450 193479}{1287 49266 157850}{2178 25415 180810}{1350 46046 161007}
418418 4669544880 :6 {12369 62920 133920}{10602 93775 104832}{19840 33759 155610}{16093 43200 149916}{11700 69696 127813}{14400 49972 144837}
425270 1398169080 :5 {1309 41310 170016}{935 68816 142884}{3468 13552 195615}{847 86940 124848}{1512 34408 176715}
428582 3920356440 :5 {11868 36400 166023}{14448 29068 170775}{7396 71760 135135}{9295 49680 155316}{12320 34830 167141}
432274 7850692080 :5 {42405 58212 115520}{45232 53865 117040}{34695 87362 94080}{35112 82080 98945}{36960 71960 107217}
434930 5224379160 :5 {12078 90272 115115}{12810 76384 128271}{11935 96096 109434}{14014 64416 139035}{15288 56265 145912}
440258 4358554200 :5 {19824 24805 175500}{12177 42952 165000}{7749 94380 118000}{7920 87084 125125}{9504 60475 150150}
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northwolves · 发表于 2025-7-15 12:22:13

这个代码哪里不对呢？

f[pp_]:=(t=Association[];Do[{m,n,p,q}=k;s={a,b,c}={m n(p^2+q^2),p q(m^2+n^2),(m q+n p) (m p-n q)};
A=m n p q c; d=ToString@pp<>"->"<>ToString@A;
If[KeyExistsQ[t,d]==False,t[d]={d}]; t[d]=Union[t[d],Sort@{s}],
{k,Values@Solve[{2 m p (m q+n p)==pp,m p>n q},{m,n,p,q},PositiveIntegers]}];Values@t);
f[108]

复制代码

{{"108->486", {36, 45, 27}, {45, 36, 27}}, {"108->432", {30, 30, 48}}}

而周长108至少有5组解： {108,<|306->{{20,37,51}},126->{{5,51,52}},234->{{15,41,52}},270->{{27,29,52}},90->{{4,51,53}}}

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