TITLE: Status of $x^3+y^3+z^3=6xyz$
QUESTION [8 upvotes]: In

Erik Dofs, Solutions of $x^3 + y^3 + z^3 = nxyz$, Acta Arithmetica 73 (1995) pp. 201–213, doi:10.4064/aa-73-3-201-213, EuDML

the author has studied the Diophantine equation
\begin{equation}
x^3+y^3+z^3=nxyz\tag{1}
\end{equation}
where $n$ is an integer. I am interested in the specific case $n=6$. The author mentions:

For a fixed $n$-value, (1) can be transformed into an elliptic curve
with a recursive solution structure derived by the "chord and tangent
process".

How does this work for the case $n=6$? Does this generate all or infinitely many solutions? What progress has been made in this specific case?

REPLY [2 votes]: We show the integer solutions of $x^3+y^3+z^3 = nxyz$ using elliptic curve.
$x^3+y^3+z^3 = nxyz\tag{1}$
Let $s=x+y+z$, then equation $(1)$ reduces to
$(-3x+3s+nx)y^2+(-3s^2+6sx+nx^2-3x^2-nxs)y+s^3-3s^2x+3sx^2=0.$
This quadratic in y must have rational solutions, so the discriminant must be square number.
Hence we obtain
$v^2 = (-6n+9+n^2)x^4+(-2n^2s+6sn)x^3+(-18s^2+n^2s^2-6ns^2)x^2+(12s^3+2ns^3)x-3s^4.$
Furthermore, let $U=s/x$ then we obtain
$$V^2 = -3U^4+(12+2n)U^3+(-18-6n+n^2)U^2+(6n-2n^2)U+9+n^2-6n\tag{2}.$$
This quartic equation is birationally equivalent to the elliptic curve below.
$$Y^2-2nYX+(-72+12n+4n^2)Y = X^3+(-6n-18)X^2+(108+12n^2-72n)X-1944+216n^2+648n-72n^3\tag{3}$$
$U = \frac{\large{-12n^2+2nX-6X+108}}{\large{Y}}$
$V = \frac{\large{-5832+60n^3X+324n^2X-972nX-8n^3Y+72n^4+3888n-1620X+216Y-3X^3+162X^2-432n^3-18n^2X^2+X^3n}}{\large{Y^2}}$
$X = \frac{\large{-6V+18-12n+2nV+2n^2+6Un-2Un^2}}{\large{U^2}}$
$Y = \frac{\large{-4Un^3+4n^3-12U^2n^2+4n^2V+24Un^2-36n^2-36Un-24nV+108n+108U^2-108+36V}}{\large{U^3}}$
If the rank of elliptic curve $(3)$ is positive, we can obtain the infinitely many rational points.
Hence we obtain the infinitely many integer solutions of equation $(1)$.
The rank and generator are obtainded using mwrank(Cremona).
Example of $n=6$.
Minimal Weierstrass form is $Y^2+Y=X^3-54X-88$.
Rank=1 and generator=$P(-2,3)$.
The rational points $(X,Y)$ of $Y^2+Y=X^3-54X-88$ are obtained by group law using Online Magma Calculator as follows.
           P<X>:=PolynomialRing(RationalField());
           E := EllipticCurve([0, 0, 1, -54, -88]);
           P :=E![-2,3];
           2*P;
           3*P;
           4*P;
           5*P
           6*P;
           7*P;

The rational points $(X,Y)$ of $Y^2+Y=X^3-54X-88$.
$1P:[   -2,     3]$
$2P:[   40,   248]$
$3P:[-143/36, 1621/216]$
$4P:[56404/5041, 9350169/357911]$
$5P:[-15323534/2505889, 12752626540/3966822287]$
$6P:[3494518273/430479504, -31008773293919/8931588748992]$
$7P:[-43215340027190/7570240479649, -132725251679433577707/20828812647389616143]$
In this way, we can obtain the infinitely many rational points of $Y^2+Y=X^3-54X-88$.
Hence there are infinitely many integer solutions for equation $(1).$
We can get the solutions using group law as follows.
$mP: [    x,      y,      z]$
$1P:[    1,     2,     3]$
$2P:[   52,   -21,   -19]$
$3P:[ 1817,  3258,  5275]$
$4P:[-2847511, 3096807, -124904]$
$5P:[10840875082, 4904676969, 15051171563]$
$6P:[-150777667094725, 458665691607396, -203863624933571]$
$7P:[81821352777652044467, 29381282043563909553, 46875396961726681714]$
The positive solutions are the case of $P,3P,5P,7P$.
We show only some solutions.
            [n] [rank] [  x     y     z]
            

            [  3][ ][    1,     1,     1]
            [  5][0][    1,     2,     1]
            [  6][1][    1,     3,     2]
            [  9][1][    3,     7,     2]
            [ 10][1][    5,    18,     7]
            [ 13][1][    9,    38,    13]
            [ 14][1][    2,    13,     7]
            [ 15][1][    7,    -1,    -3]
            [ 16][1][   70,    -9,   -31]
            [ 17][1][    5,    37,    18]
            [ 18][1][   95,    42,    13]
            [ 19][1][    9,     5,     1]
            [ 20][1][   61,   -13,   -14]
            [ 21][1][    2,    21,    13]
            [ 26][1][   91,    38,     9]
            [ 29][1][   43,   182,    27]
            [ 30][1][   31,    21,     2]
            [ 31][1][   37,    -1,   -27]
            [ 35][1][   97,   -14,   -19]
            [ 36][1][  151,    -7,   -78]
            [ 38][1][   70,   629,   151]
            [ 40][1][    9,    -1,    -2]
            [ 41][2][    1,     9,     2]
            [ 44][1][  819,   -19,  -554]
            [ 47][1][  845,   -38,  -367]
            [ 51][1][    9,    77,    13]
            [ 53][1][    2,    27,     7]
            [ 54][1][    2,    57,    43]
            [ 57][1][   91,   310,    19]
            [ 62][1][13559153, -1513300, -1950953]
            [ 63][1][ 3775,  -247,  -903]
            [ 64][1][1338039, -119479, -232736]
            [ 66][1][    3,    14,     1]
            [ 67][1][ 1133, 23517,  7525]
            [ 69][2][    2,    73,    57]
            [ 70][1][1478979, -27083, -896668]
            [ 71][1][   67,    -7,    -9]
            [ 72][1][-404512675962, 5450170263655, -1012930784383]
            [ 73][1][89200900157319, 2848691279889518, 1391526622949983]
            [ 74][1][  133,  4607,  2502]
            [ 76][2][   45,    -2,   -13]
            [ 77][2][   52,    -5,    -7]
            [ 83][1][    5,    61,     9]
            [ 84][1][   56,    -1,   -31]
            [ 86][1][    2,    91,    73]
            [ 87][1][   21,    -1,    -5]
            [ 92][1][-20446843218005, 35661385544981, -548624531286]
            [ 94][2][   19,   945,   746]
            [ 96][1][   38,    -3,    -5]
            [ 98][1][-2559169, 59978401, -14154192]
            [ 99][1][14466072543, -1832602198, -1150522313]
            [101][1][ 1271,  3078,    79]
            [102][1][459338480695732254, 3816006884967068935, 13212742329826830581]
            [103][1][   61,    -4,    -9]
            [103][1][58383, -1159, -26024]
            [105][2][    2,   111,    91]
            [106][2][    1,    54,    35]
            [107][1][-197624310994, 1329876450605, -83341950251]
            [108][1][   39,    -2,    -7]
            [109][1][-99054267227, 7254524660292, -4035385003297]
            [110][1][ 2745, 18578,  1147]
            [112][1][81634675793306734523552997069865, -66756829882613387041310733449793, -403909122691328588518061393542]
            [113][1][345842, 6313383, 15170275]
            [116][1][ 1204,   -13,  -739]
            [117][1][  545, 10318,  1677]
            [119][1][   49,    -1,   -19]
            [120][1][ 8869,  -496, -1317]
            [122][1][25590382918388481967217, 407249928739620845890, 23848086141138276680923]
            [123][1][-45191178833837, 10554611259665663, -8723981310176706]
            [124][1][-1882858151, 4389003335, -75992904]
            [126][2][    2,   133,   111]
            [127][1][  931,   -45,  -151]
            [128][1][1158179, -422318, -23611]
            [129][2][   70,  2361,   629]
            [130][1][16177096946436536530, 639905104499493910311, 1046599292363750394389]
            [132][1][ 2234,   -39,  -905]
            [133][1][691440137111968428652609, 8149000233894575265542178, 27006382877335430051053793]
            [136][2][ 1118,   -45,  -203]
            [142][1][6587432496387235561093636933115859813174, 53881756527432415186060525094013536917351, 222932371699623861287567763383948430761525]
            [143][1][1636453, -1520593, -2435]
            [145][1][157591646586434781, 44634584148027469, 1007950541819512850]
            [147][1][   21,  1529,   925]
            [148][1][1418519131294563, -188778746384314, -71841303293459]
            [149][2][   45,    14,     1]
            [151][2][  133,    -9,   -13]
            [154][2][    2,    63,    13]
            [155][1][466371,  -458, -442729]
            [156][1][-57378032205801587, 151742066509610694, -2433329851945933]
            [158][1][5642215349875, 7336556299898, 80828288788977]
            [159][1][   31,    -1,    -6]
            [160][2][ 3691,   -43, -1764]
            [161][2][   95,    -7,    -8]
            [162][1][   35,  2881,  1854]
            [164][1][-2187625242203395484208435, 9967112990856026231233891, -273965892543545308964986]
            [166][1][  790,   611,     9]
            [167][1][3641058343253213, -868179733296745, -90197542563908]
            [172][1][23593229783585883, -16590668075015195, -127237919517328]
            [174][2][   78,     7,     5]
            [175][1][-12984427575, 55614086497, -1343816497]
            [177][1][11586299300246645065650175011667633294528995894742493608006903, 499128047096078689216614212030144552460525030444760940918765730, 944421945175253160922633489847006529687358187395396664036033027]
            [178][1][    2,    97,    27]
            [181][1][201705586625136962, 10672860536839861, 21088064331923949]
            [185][1][  379,    -4,  -175]
            [186][2][  252,    -5,   -67]
            [187][1][492233837876182300994422946725623025365, -50621375726791887196233101919521352691, -25564297137318411424451907466482829394]
            [189][1][18396,  -209, -7891]
            [190][1][-297115335963207388794859858793856411, 12449186314350611078793751630598133592, -2716813894306138435285959241731689173]
            [191][1][56059, -1399, -11655]
            [192][1][ 3345,   -38, -1417]
            [195][2][    7,   143,    15]
            [196][2][  259,   -18,   -19]
            [197][1][  127, 11655,  6278]