(a+b+c)^3=kabc的正整数解

wayne

1)如果 $(a+b+c)^3=kabc$ 有整数解,那么k是哪些数
2)如果 $(a+b+c)^3=kabc$ 有正整数解,k又是哪些数

northwolves

1. Union@Select[Flatten@Table[(a + b + c)^3/(a  b  c), {a, 500}, {b, a, 500}, {c, b,500}], 0 < # < 101 && IntegerQ@# &]

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{27, 32, 36, 40, 42, 49, 50, 54, 56, 66, 72, 75, 81, 90, 96}

mathe

很显然，如果都是正整数，一个必要条件是\(k\ge 27\).
而a=-b,c=0是一组平凡解，对于所有k都成立。
通常情况，a+b+c非零时，我们可以替换为\(x=\frac a{a+b+c},y=\frac b{a+b+c}\)，转化为\(h(x,y)=kxy(1-x-y)-1=0\),这是一条三次曲线。
求\(\frac{\partial h}{\partial x}=\frac{\partial h}{\partial y}=0\),可以得到一个解x=0,y=0, 所以我们可以考虑把一条经过这个点的直线先投影到无穷远，比如我们可以选择y轴，于是做变换\(X=\frac 1x, Y=\frac yx\),得到\(X^3=k(YX-Y^2-Y)\).
此后我们需要消除YX-Y，为此我们需要做替换\(Z=Y-\frac X2+\frac 12\),或\(Y=Z+\frac X2 -\frac12\), 变换后得到
\(-\frac{kX^2}4 + \frac{kX}2 + k Z^2 - \frac k4 + X^3=0\)
这时离标准化已经不远了，两边可以同时乘上\(k^3\)变为
\(-\frac{k^4X^2}4 + \frac{k^4X}2 + k^4 Z^2 - \frac {k^4}4 + X^3k^3=0\)
取\(V=8k^2Z, U=-4kX\)得到，
\(V^2=U^3+k^2 U^2+8k^3 U +16k^4\).
曲线上有有理点\(U=0,V=\pm 4k^2\)是三阶点。
可以计算得到上面复合变换实际上是
\(U=-\frac{4k}{x}, V=\frac{4k^2(x+2y-1)}x\)
逆变换
\(x=-\frac{4k}{U},  y=\frac{4k^2+kU-V}{2kU}\)
题目中有正整数解，等价于\(x\ge 0, y\ge 0, x+y\lt 1\)
所以在\(k\ge 0\)时，我们需要\(U\lt 0, 4k^2+kU-V\lt 0, 4k^2+kU+V\lt 0\)
如下图(对应k=32), 不等式边界分别对应曲线在\((0,\pm 4k^2)\)处两条切线，分别要求是绿色直线上方和红色直线下方，所以只能是椭圆曲线左边封闭部分。

也就是我们需要找到曲线左边封闭部分的有理点，会正好对应原方程的正整数解。
然后比较有意思，由于右边分支上任意两点连线交出第三点也必须在右边，所以如果有符合条件的，要么椭圆曲线生成元中出现，要么有限阶点中出现，这个使得判断变得比较容易了。
进一步观察函数\(f(U)=U^3+k^2 U^2 + 8k^3 U+16k^4\)
我们可以发现\(f(0)=16k^4\gt 0, f(-4k)=-64k^3\lt 0, f(-8k)=16k^3(k-32)\),
所以在\(k\ge 32\)时，我们知道函数\(f(U)\)在区间(-4k,0)有一个根，在 [-8k, -4k)有一个根，在\((-\infty, -8k)\)有一个根。
由此对于曲线上的一个候选点，我们只需要判断这个解的U坐标是不是小于-4k就可以判断它是不是在左边封闭区间了。 （而\(k\lt 32\)应该只有\(k=27\)有解，这时椭圆曲线退化了。
比如k=31,可以得到f(U)三个根为-610.**， -256.**， -94.***，
而它有三阶点[0,3844], 和生成元[156, 8896], 显然两者都在右分支，所以k=31没有整数解。
而比如k=32, 曲线rank=0, 有六阶点[-512, 4096] ，其中-512<4*32, 所以在左分支。所以具有有限个本原正整数解。
k=33, 三阶元[0,4356]显然在右分支，生成元[-512, 4096]，其中-512<4*33, 所以在左分支，具有无限个本原正整数解。

下面就可以尝试去尝试两个序列了：具有正整数解的k和具有无限个正整数解的k，用pair/gp计算结果如下
32 [1, 0]
33 [0, 1]
36 [0, 1]
37 [0, 1]
40 [0, 1]
41 [0, 1]
42 [0, 1]
43 [0, 1]
49 [0, 1]
50 [0, 1]
54 [1, 0]
56 [0, 1]
62 [0, 1]
65 [0, 1]
66 [0, 1]
68 [0, 1]
72 [0, 1]
73 [0, 1]
75 [0, 1]
78 [0, 1]
81 [0, 1]
82 [0, 1]
其中[1,0]代表只有符合条件有限阶点，而第二项为1代表有符合条件无限阶点。不过对于85,86, 我的电脑上用pari/gp求generators会报错，所以就无法继续算下去了。

wayne

我用椭圆曲线搜了下, 有正整数解, 1000以内的有(可能会漏掉,因为部分曲线出现了问题)

1. 32,33,36,37,40,41,42,43,49,50,54,56,62,65,66,68,72,73,75,78,81,82,86,88,90,91,96,101,104,113,118,122,125,126,132,133,136,138,142,145,146,149,150,152,153,157,158,160,161,165,169,170,173,174,177,180,182,185,189,196,198,201,203,205,216,219,221,225,230,237,242,245,246,250,254,260,261,267,270,274,285,286,294,296,308,312,314,318,322,325,326,330,341,342,344,346,350,354,357,361,365,366,369,372,376,377,378,382,388,389,392,393,402,405,406,414,425,427,434,438,442,446,449,450,456,458,459,470,473,475,477,480,482,484,486,488,498,500,504,509,525,532,534,544,545,546,549,550,554,555,566,570,585,586,596,597,600,605,608,610,619,625,629,630,632,637,642,648,651,657,677,680,682,689,693,702,706,713,726,729,734,741,749,756,761,765,770,776,782,785,792,800,801,806,834,841,845,854,864,865,870,875,882,889,897,909,914,916,918,933,936,952,953,957,961,966,973,980,984,987,992

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放在了https://nestwhile.com/res/emath/50005/positive

sageMath不能确定所有生成元的还挺多的,需要借助magma进一步求解,这都是手工输入在线magma的过程,我就暂时不折腾了

1. {109,113,118,142,149,158,163,166,193,197,206,229,232,239,251,253,262,268,269,277,278,282,283,298,301,305,349,353,356,367,373,379,394,398,403,408,409,413,417,418,419,421,422,424,430,443,445,478,481,485,487,489,493,499,503,505,524,537,538,541,542,543,548,553,562,563,565,568,569,571,573,577,581,584,587,589,592,593,595,598,604,607,613,618,635,638,641,643,647,649,653,654,659,661,662,668,669,688,691,694,697,698,699,716,721,724,730,733,738,739,742,743,745,753,758,760,763,764,769,772,773,786,788,790,793,794,795,797,798,804,805,807,809,814,818,821,829,830,835,837,839,842,843,849,856,858,862,863,869,873,877,878,883,885,888,893,895,898,899,901,902,904,906,907,908,911,913,917,922,923,926,928,934,937,938,941,942,949,955,958,964,967,977,982,983,985,993}

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只有负整数解的是

1. {2,7,10,12,14,15,19,22,31,44,51,53,61,67,76,80,83,85,87,89,111,114,115,116,123,124,127,131,140,159,164,167,172,175,176,179,183,184,186,187,199,208,210,211,214,220,231,233,241,247,252,255,256,259,279,281,284,287,290,303,304,306,323,329,335,340,351,355,362,364,371,380,385,391,396,399,407,411,412,428,436,439,444,451,452,454,460,471,474,476,492,495,511,515,519,528,531,536,559,567,575,576,588,591,599,606,611,615,620,624,627,628,636,644,663,667,671,692,701,711,714,717,722,728,732,735,751,752,759,766,767,768,783,784,796,803,810,815,824,826,828,831,836,848,850,851,855,879,880,884,887,896,900,903,905,912,924,927,931,939,945,954,956,960,963,970,976,978,986,988,995,996,999}

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https://nestwhile.com/res/emath/50005/all/
我发现了一个很明显的规律就是 三个数都是立方数或者平方数的概率 非常的高

.

wayne

可惜OEIS上没有收录这个数列.倒是收录了$(x^3 + y^3 + z^3)=kx*y*z$的正整数解的数列 https://oeis.org/A072716/

wayne

我发现一个特别有意思的地方, 就是$k$的一个素因子不可能同时分布在$(a,b,c)$里的两个数里.当然这个很容易证明.,因为如果两个都含有k的因子$k_1$,必然第三个也能被$k_1$整除.这个跟解的互质性矛盾了.
这个对于之前的问题$(a+b+c)^2=kabc$同样适用.

这就意味着 $k$的素因子只有一个的时候,比如$k=p,p^2$, 那么解$(a,b,c)$都是形如$(p_1^3,p_2^3, k*p_3^3)$,其中$p_1,p_2,p_3$不含k的素因子p.
而椭圆曲线可以轻轻松松产生 几千,几十万,几百万位的整数, 随便一个解 Mathematica都很难分解,但是我们可以确定肯定形如$(p_1^3,p_2^3, k*p_3^3)$

比如$k=43$, https://nestwhile.com/res/emath/50005/positive/a-043-1.txt ,  秩为1. 第4个解分别是{490, 491, 491} 位,开三次方后,Mathematica竟然还是不能分解

比如$k=19^2=361$,https://nestwhile.com/res/emath/50005/positive/a-361-1.txt, 几个解也都是$(p_1^3,p_2^3, 19p_3^3)$,其中$p_1,p_2,p_3$不含k的素因子19
比如$k=59*2=118$,https://nestwhile.com/res/emath/50005/positive/a-118-2.txt, 几个解也都是$(p_1^3,p_2^3, 118^2p_3^3)$,其中$p_1,p_2,p_3$不含k的素因子2和59

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如此说来,本题$(a+b+c)^3=kabc$ 等价于讨论$A x^3+B y^3+C z^3 =n xyz $的正整数解.其中$n^3=kABC$, 其中$A,B,C$两两互质. 更强的是 $gcd(A x^3,B y^3)=1,gcd(A x^3,C z^3)=1,gcd(B y^3,C z^3)=1$
...

wayne

当$A,B,C$至少有一个为1的情况,比较特殊,很容易构造基本解.

但是当$A,B,C$都不为1,就不那么显然了.
进一步统计发现1000以内,k为以下数的时候,有三个以上的素因子,且存在 三个素因子都分散到 不同的数的解如下

1. 66,78,90,126,132,150,165,180,230,285,308,318,322,330,357,450,470,525,534,546,555,610,630,642,726,765,770,782,854,870,897

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比如$k=66=2*3*11$的时候, $4 x^3+9 y^3+121 z^3 =66 xyz$ 有特解$(2,3,1)$, 等价于 椭圆曲线$Y^2 =X^3 -1689039*X - 815123034$,正变换是$[x,y]=[\frac{2 X^3+10197 X^2+111 X Y+8213238 X+234135 Y+503225811}{X^3+855 X^2+2041875 X+1432197261},\frac{3 \left(X^3+1617 X^2+50 X Y-488961 X-2178 Y-624189753\right)}{X^3+855 X^2+2041875 X+1432197261}]$,逆变换是$[X,Y]=[\frac{3 \left(95 x^2-450 x y-638 x-333 y^2-99 y+2783\right)}{(x-2)^2},-\frac{27 \left(1266 x^3-762 x^2 y+1078 x^2-2829 x y^2-10989 x y-6655 x+1551 y^2+12705 y+22143\right)}{(x-2)^3}]$
其他的解是

1. (1663742860803843466, 668001740979696153, 521006325057295643)
   
2. (62177810153309616118252400027711189824780566080151074619555503750532402922760732888357738, 62726042619580919270344072566191061461607876323993528598894314593742876353872925026679659, 13528180092747888863903947570390641012377302581286448440451014745251392701025691217887273)
   
3. (215683020745800959194895345848556452858408635149372092991334974656880567659090357749482848904683570490963280621079886779852772770969894786445199899634094775326766140611251227419586093396669751378085955618348184098, 172196444174237652838185253449718700284062592626067351548938439061267016028381563632955109502944193992275455569533880665353538123583966171135136296869049603369067233353751070819702987974632686445787267245088634737, 119627833771479383885695111615196755014892050892011635293541175600643262706586170365949767255904770866261586843511754490736487540593187922592613204901075388662239897744379593544252484521545373042971281512946249779)
   
4. (154790843622724743837511301529329347717526537711457184148753315449378601955222995736522443756351247289794102711090163929781182720080620827236721322688998871765786605279830046437451775232074208789942467676955996841709444538608206954502641046865974439119227653358885293113376298108148600995131496087582756816917708178133586983409622824194573869895349961805080998563908024256422490314499068114, 84261182509619887077500505801564306459901489544412734526227138694626359681572778050840454746355553075504734323439341072990280684849039217388645053785280521366539645726167296490734329452305472553497334433343119971241404392490224753936097967600170752790048194959830681225946824138321106862227540746923388613210078556670930256676987977997214681402980686377266158820271815329610476614113745747, 25942660420784236212081931321063396329237029470554399328566232578470881973494126612811015254170502674919261766749787535929050252822606626398213989699065189277197058445143842655225020430806542901682730257086578883377085633801252712757723676881935267976344250367310671407904406529171827124469724545579012432226705489153914867833814292082362571860704369220815555986528386506975397916562975697)
   
5. (3033064669931161092936922130520344018865537595513247970912925579609664879434618230439543773306888289644012537439557947717618430417188376861240596841152958303242174032206325881231292484579336106060187916657981749803125234894016541947814630965307601367229038585101684629839319762643618543172952688837887609054047329691872550737803003851714935238095104300725268953238681404835623031939849377646004421857308681553603147906794796846043540680657599625305794473207334854271090989951413587205933047579950660918568330288977449263169853087956776944622090197989163155570610851598592067612618589992847993754829109282155685348540602, 4498559473796079116011844471563217589150983999468207860270026319643590289886076598113174958160584867391629265966782118115834869343935098649654399837938112579086282500186865872823649711793647421446041098648262463522163189573117017933336058701815743819374633677803586735680382787592573889887561756391309993259852916582816667081559233131193148800270668049989736963655048575605399044654714006825639509208033626541580102246723575060197526325624147426324292940238370630919433169299343593396972789332904671710046439042590329581035380072259268752045579486304156415132321294901883903638788000781834308221745382047053341612505161, 1732497564569226553203672946316290585792177415822833372987159857715878957208034649287373809189448013645101361244622615392023617489661113297381984166368510675569368896758561716662639823010753730101774414340001507136879046820023411654130646109966386185378431441189460859161727882757648808222363864132130111881867147608160936466426582673567468209663587282021644885687289146191151223092162358614806547792055831416876673191368463795807886327870635467817285537452660102987434543830515615942198241033800779221664399945330894921455172288422941759264814248496746539389517076416269205380847996391485988283008866983295702137671627)
   
6. (24572190935793130418077293991292157028822541118332120677507456587968654130848261698472218272023390058390960261460359247493110169842458832229563760520976306980505668885888565944413448301847037243693726784391170252591865581795085910377193806254059596366881291207008530661600815208624642375805943613219073308332088448498895392655660914106776414665177875823022072203036073508484788947503785311044467871513873304559266411751823241473320984122726842648668825532021129420853110017475636133492744551995639394825984052837690973097926325386014991843058731700344722006191324186836336377161309203884215193435366141517762483932916962068447427687722724054631915371692813531725513008884300602186202087881858832872628121730720381936140331836303773257722038896254077017109194632861630923576362215612629415496685837424509068997833719508097541404186557896303444936590084008589396375027494559118065172951324779357979039802, 9522775980471025709535447172250883861986255905033258701401243935161168605589990462817862873317540343344802367884395422173954394314454374286627243124297888687502959990685699179302461320533503338082493951476926886197643712284945075175943392448523353278648322169799817693083857262292460797584764339000277604911693126234021458971745795822861680158523530402664710477237773242025975342976649068604013344872827075296186918266741896984058059190861692656685805159009298863256433802400657418876865262065935458852324757840436085715538394721470070552489200574298923193385673248504424993591179707863674864627811796176686883690477777568866968696701373832406585229385915183494490237478098086842993149916250187914607967920810739791449297841686200509683147225163564927610081092851070654917787434366690185920233837310853435302499260141253691816743817351049520465456468944908573630403415261343161895316879456411998574523, 6648254877892295003409574196463626505940103043174136921070290188863360544694290998679690369827657688804366876399963822333278324023609554722908731732621536091908763490318286364530722460193332463972538864228188428753800450743143688364354294989262284073504226635791983255868858135406789617450196884434118123083401451059042356388061377748371864837945930636648299096128609992040633611386381632849671368484484711341384610014942193617419104450356015015737058059702116041947567607584025042286085355729149213366071916128466365571281051072098799780836336325844850679209937774945288789057024915022197321466440910391799701707003528321628513446720207867316166328217274504291954125165566439227088955168250422849057992501137080463337232564660947084966653807283858195759306159800146197216905569232312393564557539627063357004959465186390311168787584966281434839143385893479716474212878202052314769298035794074425314041)
   
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