周长、面积相等的本原三角形

mathe · 发表于 2025-7-28 08:08:14

第二组8个三角形的:
3255514 229793644080 :8 {92036 280896 1254825}{122265 203840 1301652}{73568 373464 1180725}{125685 197912 1304160}{117325 213136 1297296}{56628 575225 995904}{56784 571725 999248}{63360 466235 1098162}

wayne · 发表于 2025-7-28 15:33:54

简单计算了一下, mathe的这个数据在我那里,需要搜索到$[u,v]=[55/388, 2660/1067]$, 这个已经超纲了. 500以内的我都没有搜索完.更别提1000以外的范围了.
看来有理参数解的方式搜索也有很大的局限性.

wayne · 发表于 2025-8-18 23:33:38

northwolves 发表于 2025-7-11 23:00
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}}}

原则上,等腰三角形也是本原解,不能排除,所以,3组解最小的是98. 而不是[210, 840] (35, 73, 102) (25, 84, 101) (21, 89, 100)

3: [98, 420] (29, 29, 40) (25, 34, 39) (24, 37, 37)

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wayne · 发表于 2025-8-19 17:51:10

混杂有等腰三角形的本原解也挺多的. 我给出几组,4,5的.
周长和面积的关系是 $2 m^2 = p, m n (m^2 - n^2) = 2 S$,其中$gcd(m,n)=1$.

{5,{935/2341,935/2341},{10960562,5040948292380},{{3177253,3177253,4606056},{2696344,3742237,4521981},{2619071,3863666,4477825},{2495761,4172590,4292211},{2490840,4234861,4234861}}}
{4,{7/37,7/37},{2738,170940},{{709,709,1320},{445,984,1309},{319,1138,1281},{280,1229,1229}}}
{4,{69/139,69/139},{38642,69822480},{{12041,12041,14560},{11677,12444,14521},{11271,13042,14329},{11040,13801,13801}}}
{4,{91/139,91/139},{38642,69822480},{{12041,12041,14560},{11677,12444,14521},{11271,13042,14329},{11040,13801,13801}}}
{4,{29/211,29/211},{89042,133638960},{{22681,22681,43680},{8996,37033,43013},{7575,38866,42601},{6496,41273,41273}}}
{4,{203/643,203/643},{826898,24294057480},{{227329,227329,372240},{158249,308498,360151},{150249,323491,353158},{146160,340369,340369}}}
{4,{133/677,133/677},{916658,19837833120},{{238009,238009,440640},{148504,331713,436441},{111325,377689,427644},{104304,389017,423337}}}
{4,{177/877,177/877},{1538258,57263978100},{{400229,400229,737800},{190929,631954,715375},{177359,654770,706129},{168504,684877,684877}}}
{4,{209/1049,209/1049},{2200802,115838174760},{{572041,572041,1056720},{333761,824236,1042805},{272156,902725,1025921},{249771,940006,1011025}}}
{4,{495/1549,495/1549},{4798802,825939284940},{{1322213,1322213,2154376},{1019701,1657429,2121672},{929071,1785106,2084625},{859320,1969741,1969741}}}
{4,{525/2189,525/2189},{9583442,2595005212800},{{2533673,2533673,4516096},{1951520,3139721,4492201},{1650089,3483596,4449757},{1617721,3522985,4442736}}}
{4,{285/2263,285/2263},{10242338,1625268541260},{{2601197,2601197,5039944},{813775,4500249,4928314},{799239,4520674,4922425},{681720,4780309,4780309}}}
{4,{3059/4819,3059/4819},{46445522,102196271617440},{{14299561,15225177,16920784},{14072464,15587497,16785561},{13888745,16073476,16483301},{13865280,16290121,16290121}}}
{4,{2635/5659,2635/5659},{64048562,186997644353520},{{19483753,19483753,25081056},{18799641,20218396,25030525},{17630041,21850546,24567975},{17116960,23465801,23465801}}}
{4,{4797/5875,4797/5875},{69031250,162110882934000},{{16470377,21218341,31342532},{16437257,21254216,31339777},{13326457,24942841,30761952},{11504416,28763417,28763417}}}
{4,{15257/17143,15257/17143},{587764898,7991212405453200},{{132271109,172707340,282786449},{91516945,216185629,280062324},{70830657,241732417,275201824},{61106400,263329249,263329249}}}
{4,{19809/21041,19809/21041},{885447362,10488217896248400},{{86587681,365127085,433732596},{82088056,370125897,433233409},{50459105,415730431,419257826},{50327200,417560081,417560081}}}

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mathe · 发表于 2025-8-20 16:29:12

由于a+b-c,b+c-a,c+a-b同奇偶，如果它们都是奇数，那么$x=\frac{b+c-a}2,y=\frac{c+a-b}2,z=\frac{a+b-c}2$全部是半整数，得到面积平方$S^2=xyz(x+y+z)$必然不是整数，所以面积整数时x,y,z都是整数。
如果有x=z, 那么我们有$2x+y=p, x^2y=\frac{S^2}{p}=H$,
消去y,我们得到x满足方程$x^3-\frac{p}2x^2+\frac H2=0$。
由此我们得到对于给定周长和面积，x最多有三个不同的解，而且它们乘积为$-\frac H2 \lt 0$,所以其中至少一个负数，所以x最多两个正整数解。也就是最多两组等腰Heron三角形面接和周长都相同。
现在我们假设有两个等腰Heron三角形面积和周长都相同，它们对应的x为$x_1,x_2$, 记上面方程第三个负根为$-x_3=\frac{p}2 -x_1-x_2 \lt 0$时整数或半整数。
根据韦达定理有$x_1x_2-x_3(x_1+x_2)=0$, 也就是$x_1+x_2|2x_1x_2$.
设$g=gcd(x_1,x_2), x_1=g u, x_2=gv$, 于是$(u,v)=1$,而且$u+v|2guv$,所以$u+v|2g$, 所以我们设$g=\frac{(u+v)d}2$,得到
$x_1=\frac{(u+v)ud}2,x_2=\frac{(u+v)vd}2,x_3=\frac{uvd}2,p=du^2 + dvu + dv^2, y_1=dv^2,y_2=du^2$
由于我们要求$(x_1,y_1)=1,(x_2,y_2)=1$,所以只能选择d=1或2.
如果d=2,我们必须要求$x_1,x_2$都是奇数，于是$(u+v)u,(u+v)v$都是奇数，这是不可能的，所以可以得到d=1,u,v都是奇数。
于是通解为，u,v为任意互素的奇数，$x_1=\frac{(u+v)u}2,x_2=\frac{(u+v)v}2,p=u^2 + vu + v^2, y_1=v^2,y_2=u^2, H=(\frac{uv(u+v)}2)^2$
由于$H=\frac{S^2}p$,所以如果S是整数，那么$p=u^2+uv+v^2$必然是完全平方数.
由此，我们可以要求$u=s^2-t^2, v=2st+t^2$,其中t为奇数，s为偶数，而且(s,t)=1。

mathe · 发表于 2025-8-20 16:59:16

反之，我们假设有一个等腰Heron三角形，对应x=z, 于是我们有$2x+y=p, x^2yp=S^2$
由此我们得到$y(2x+y)=(\frac{S}x)^2$
由于(x,y)=1, 所以$gcd(y,2x+y)|2$.
所以只有两种可能
i) $gcd(y,2x+y)=1, y=v^2, 2x+y=w^2,S=xvw, (v,w)=1$
对于可以得到$2x=w^2-v^2, p=w^2,S=\frac{vw(w^2-v^2)}2$,这里只需要选择v,w都是奇数就可以了。
ii)$gcd(y,2x+y)=2, y=2v^2, 2x+y=2w^2,S=2(w^2-v^2)vw, (v,w)=1, x=w^2-v^2$,这里要求(v,w)=1而且两者奇偶不同。
当然这样得到的三角形不一定存在周长面积相同的其它本原Heron三角形

wayne · 发表于 2025-8-20 17:12:02

wayne 发表于 2025-7-14 11:49
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} ...

利用这里的结论,代入$u=v$,即$q=m,p=n$得到三边是$(m^2+n^2,m^2+n^2,2(m^2-n^2))$,消除公约数$mn$,得到周长$dP=4m^2,d^2S=2mn(m^2-n^2)$
由于要求$(m,n)=1$,所以分两种情况
1) 如果m,n都是奇数,就有周长$P=2m^2,2S=mn(m^2-n^2)$
2) 如果一奇一偶,可以得到周长$P=4m^2,S=2mn(m^2-n^2)$

wayne · 发表于 2025-8-20 17:59:06

搜了下,找到了周长$P=4m^2$的形式.虽然不多,占比非常的少. 不到1/100.

{3,{2/37,2/37},{5476,202020},{{1373,1373,2730},{533,2218,2725},{203,2570,2703}}}
{3,{10/109,10/109},{47524,25682580},{{11981,11981,23562},{3434,20787,23303},{2937,21386,23201}}}
{3,{195/3044,195/3044},{37063744,10955006822760},{{9303961,9303961,18455822},{3791183,14860347,18412214},{1394323,17505847,18163574}}}
{3,{253/512,253/512},{1048576,51331230720},{{326153,326153,396270},{321750,330743,396083},{304937,353513,390126}}}
{3,{3741/10364,3741/10364},{429649984,7243908462346920},{{121407577,121407577,186834830},{109758862,133737097,186154025},{89789375,162058187,177802422}}}

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大部分都是周长$P=2m^2$的形式

{3,{5/7,5/7},{98,420},{{29,29,40},{25,34,39},{24,37,37}}}
{3,{3/7,3/7},{98,420},{{29,29,40},{25,34,39},{24,37,37}}}
{3,{25/31,25/31},{1922,130200},{{471,586,865},{436,625,861},{336,793,793}}}
{3,{2/37,2/37},{5476,202020},{{1373,1373,2730},{533,2218,2725},{203,2570,2703}}}
{4,{7/37,7/37},{2738,170940},{{709,709,1320},{445,984,1309},{319,1138,1281},{280,1229,1229}}}
{3,{35/43,35/43},{3698,469560},{{1009,1009,1680},{679,1394,1625},{624,1537,1537}}}
{3,{13/43,13/43},{3698,469560},{{1009,1009,1680},{679,1394,1625},{624,1537,1537}}}
{3,{35/53,35/53},{5618,1469160},{{1731,1753,2134},{1599,1945,2074},{1584,2017,2017}}}
{3,{35/67,35/67},{8978,3827040},{{2857,2857,3264},{2725,3044,3209},{2704,3129,3145}}}
{3,{17/73,17/73},{10658,3127320},{{2809,2809,5040},{1521,4258,4879},{1360,4649,4649}}}
{3,{11/91,11/91},{16562,4084080},{{4201,4201,8160},{1241,7346,7975},{1056,7753,7753}}}
{3,{57/97,57/97},{18818,17029320},{{6217,6217,6384},{6175,6274,6369},{6160,6329,6329}}}
{3,{55/97,55/97},{18818,17029320},{{6217,6217,6384},{6175,6274,6369},{6160,6329,6329}}}
{3,{10/109,10/109},{47524,25682580},{{11981,11981,23562},{3434,20787,23303},{2937,21386,23201}}}
{3,{85/113,85/113},{25538,26625060},{{7225,7414,10899},{6154,8655,10729},{5544,9997,9997}}}
{4,{69/139,69/139},{38642,69822480},{{12041,12041,14560},{11677,12444,14521},{11271,13042,14329},{11040,13801,13801}}}
{4,{91/139,91/139},{38642,69822480},{{12041,12041,14560},{11677,12444,14521},{11271,13042,14329},{11040,13801,13801}}}
{3,{115/151,115/151},{45602,83143620},{{10201,16303,19098},{10166,16361,19075},{9576,18013,18013}}}
{3,{25/157,25/157},{49298,47147100},{{12637,12637,24024},{4849,21074,23375},{4200,22549,22549}}}
{3,{15/169,15/169},{57122,35915880},{{14393,14393,28336},{3151,26146,27825},{2640,27241,27241}}}
{3,{135/173,135/173},{59858,136673460},{{13000,21109,25749},{11879,23153,24826},{11704,24077,24077}}}
{4,{29/211,29/211},{89042,133638960},{{22681,22681,43680},{8996,37033,43013},{7575,38866,42601},{6496,41273,41273}}}
{3,{119/229,119/229},{104882,521584140},{{33301,33301,38280},{31871,35186,37825},{31416,36733,36733}}}
{3,{145/229,145/229},{104882,521584140},{{33301,33301,38280},{31871,35186,37825},{31416,36733,36733}}}
{3,{93/247,93/247},{122018,601380780},{{34829,34829,52360},{27591,43618,50809},{26040,47989,47989}}}
{3,{19/271,19/271},{146882,188144460},{{36901,36901,73080},{6409,68482,71991},{5320,70781,70781}}}
{3,{119/271,119/271},{146882,955860360},{{43801,43801,59280},{42756,44881,59245},{40495,47506,58881}}}
{3,{95/277,95/277},{153458,890815380},{{42877,42877,67704},{31775,56114,65569},{29640,61909,61909}}}
{3,{253/307,253/307},{188498,1174385520},{{50689,51400,86409},{35001,69049,84448},{30240,79129,79129}}}
{3,{105/313,105/313},{195938,1428707280},{{54497,54497,86944},{39729,72034,84175},{36960,79489,79489}}}
{3,{21/331,21/331},{219122,379246560},{{55001,55001,109120},{15481,95041,108600},{8656,102841,107625}}}
{3,{9/331,9/331},{219122,163070460},{{54821,54821,109480},{25105,84571,109446},{14341,95400,109381}}}
{3,{37/343,37/343},{235298,737854740},{{59509,59509,116280},{15751,105698,113849},{13320,110989,110989}}}
{3,{185/361,185/361},{260642,3208885680},{{82273,82273,96096},{78225,87586,94831},{76960,91841,91841}}}
{3,{315/367,315/367},{269378,2049907860},{{47238,96889,125251},{39550,106969,122859},{35464,116957,116957}}}
{3,{259/379,259/379},{287282,3757603080},{{84841,89917,112524},{78884,98057,110341},{76560,105361,105361}}}
{3,{23/397,23/397},{315218,717148740},{{79069,79069,157080},{11375,148754,155089},{9384,152917,152917}}}
{3,{115/403,115/403},{324818,3456966240},{{87817,87817,149184},{56425,124114,144279},{51520,136649,136649}}}
{3,{295/419,295/419},{351122,5471746140},{{103486,106177,141459},{88861,128472,133789},{88536,131293,131293}}}
{3,{41/421,41/421},{354482,1515170580},{{89461,89461,175560},{21441,160882,172159},{18040,168221,168221}}}
{3,{333/437,333/437},{381938,5826660840},{{106794,110383,164761},{85556,136213,160169},{80080,150929,150929}}}
{3,{49/439,49/439},{385442,2046986760},{{97561,97561,190320},{85705,109456,190281},{25337,174521,185584}}}
{3,{205/439,205/439},{385442,6780943260},{{117373,117373,150696},{106375,131426,147641},{103320,141061,141061}}}
{3,{51/469,51/469},{439922,2599516920},{{111281,111281,217360},{64921,158251,216750},{41497,183139,215286}}}
{3,{129/469,129/469},{439922,6150531660},{{118301,118301,203320},{73761,169522,196639},{67080,186421,186421}}}
{3,{175/481,175/481},{462722,8448476400},{{130993,130993,200736},{101311,166786,194625},{95200,183761,183761}}}
{3,{377/487,377/487},{474338,8724624480},{{125200,142129,207009},{103321,169048,201969},{95040,189649,189649}}}
{3,{301/491,301/491},{482162,11119794840},{{153961,157876,170325},{152636,159725,169801},{150480,165841,165841}}}
{3,{275/499,275/499},{498002,11895760800},{{162313,162313,173376},{159401,166226,172375},{158400,169801,169801}}}
{3,{301/499,301/499},{498002,11895760800},{{162313,162313,173376},{159401,166226,172375},{158400,169801,169801}}}
{3,{85/503,85/503},{506018,5254247460},{{130117,130117,245784},{105721,154834,245463},{47209,224211,234598}}}
{3,{377/503,377/503},{506018,10513142640},{{140721,150088,215209},{122329,171889,211800},{110880,197569,197569}}}
{3,{253/512,253/512},{1048576,51331230720},{{326153,326153,396270},{321750,330743,396083},{304937,353513,390126}}}
{3,{445/521,445/521},{542882,8510566260},{{112741,176389,253752},{93886,198025,250971},{73416,234733,234733}}}
{3,{27/547,27/547},{598418,2204125560},{{149969,149969,298480},{18409,284818,295191},{15120,291649,291649}}}
{3,{135/553,135/553},{611618,10734791760},{{162017,162017,287584},{91375,241954,278289},{82080,264769,264769}}}
{3,{141/559,141/559},{624962,11531219700},{{166181,166181,292600},{96375,245506,283081},{86856,269053,269053}}}
{3,{329/589,329/589},{693842,23125824540},{{227581,227581,238680},{224721,231442,237679},{223720,235061,235061}}}
{3,{351/589,351/589},{693842,23125824540},{{227581,227581,238680},{224721,231442,237679},{223720,235061,235061}}}
{3,{49/601,49/601},{722402,5283150600},{{181801,181801,358800},{36625,333026,352751},{30576,345913,345913}}}
{3,{245/619,245/619},{766322,24502595040},{{221593,221593,323136},{181575,270706,314041},{172480,296921,296921}}}
{4,{203/643,203/643},{826898,24294057480},{{227329,227329,372240},{158249,308498,360151},{150249,323491,353158},{146160,340369,340369}}}
{3,{209/659,209/659},{868562,26898864300},{{238981,238981,390600},{201531,278785,388246},{154506,353431,360625}}}
{3,{319/661,319/661},{873842,35335751220},{{269341,269341,335160},{248479,296242,329121},{242440,315701,315701}}}
{4,{133/677,133/677},{916658,19837833120},{{238009,238009,440640},{148504,331713,436441},{111325,377689,427644},{104304,389017,423337}}}
{3,{637/683,637/683},{932978,13208755560},{{168961,305328,458689},{89189,390313,453476},{60720,436129,436129}}}
{3,{53/703,53/703},{988418,9154536300},{{248509,248509,491400},{46359,458434,483625},{38584,474917,474917}}}
{3,{621/709,621/709},{1005362,25765712280},{{245725,280041,479596},{195286,332665,477411},{117040,444161,444161}}}
{3,{31/721,31/721},{1039682,5798743440},{{260401,260401,518880},{27871,497986,513825},{22816,508433,508433}}}
{3,{265/721,265/721},{1039682,42952940520},{{295033,295033,449616},{229825,373826,436031},{216240,411721,411721}}}
{3,{155/727,155/727},{1057058,28425016620},{{276277,276277,504504},{138425,429794,488839},{122760,467149,467149}}}
{3,{475/757,475/757},{1146098,62462492400},{{352149,383524,410425},{347449,398325,400324},{347424,399337,399337}}}
{3,{165/763,165/763},{1164338,34932337440},{{304697,304697,554944},{154599,472114,537625},{137280,513529,513529}}}
{3,{403/763,403/763},{1164338,64535791320},{{372289,372289,419760},{358969,389938,415431},{354640,404849,404849}}}
{3,{477/763,477/763},{1164338,64535791320},{{372289,372289,419760},{358969,389938,415431},{354640,404849,404849}}}
{3,{385/793,385/793},{1257698,73368455160},{{388537,388537,480624},{359569,425954,472175},{351120,453289,453289}}}
{3,{13/827,13/827},{1367858,3675551880},{{342049,342049,683760},{23269,662025,682564},{11257,676779,679822}}}
{3,{231/829,231/829},{1374482,60693693060},{{370301,370301,633880},{233359,528082,613041},{212520,580981,580981}}}
{3,{533/853,533/853},{1455218,100822962240},{{456394,475705,523119},{450449,484520,520249},{443520,505849,505849}}}
{3,{585/857,585/857},{1468898,98319770640},{{440689,452895,575314},{392562,532849,543487},{392224,538337,538337}}}
{4,{177/877,177/877},{1538258,57263978100},{{400229,400229,737800},{190929,631954,715375},{177359,654770,706129},{168504,684877,684877}}}
{3,{413/907,413/907},{1645298,122131649640},{{496609,496609,652080},{443751,563698,637849},{429520,607889,607889}}}
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{3,{109/2971,109/2971},{17653682,1427313915720},{{4419361,4419361,8814960},{403975,8509106,8740601},{329616,8662033,8662033}}}
{3,{63/2977,63/2977},{17725058,830715893280},{{4433249,4433249,8858560},{234175,8676994,8813889},{189504,8767777,8767777}}}
{3,{315/2983,315/2983},{17796578,4133998246140},{{4498757,4498757,8799064},{1166425,8012194,8617959},{985320,8405629,8405629}}}
{3,{545/2989,545/2989},{17868242,7034936284740},{{4615573,4615573,8637096},{2007825,7474066,8386351},{1757080,8055581,8055581}}}

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