Status of $x^3+y^3+z^3=6xyz$

Question

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?

Answer 1

(Collecting comments into a community wiki answer.)

There is a standard method for transforming a smooth cubic into Weierstrass form. See for example Section 1.3 or Appendix B of Silverman and Tate's book Rational Points on Elliptic Curves. It is also implemented in Sage as $\mathtt{EllipticCurve\_from\_cubic}$.

For $n=6$ your curve is isomorphic to $Y^2 + Y = X^3 -54X - 88$. According to the LMFDB, it has torsion $\mathbb{Z}/3\mathbb{Z}$ and rank 1. Thus one can start with the obvious solution $(x:y:z) = (1:2:3)$ and its permutations and generate all rational solutions. Since your equation is homogeneous, finding all rational solutions is equivalent to finding all integer solutions.

The positive solutions constitute one of two connected components of the real locus of the elliptic curve. Since the generator is on that component (and the identity is not), its odd multiples again yield positive solutions. The next batches of positive solutions are permutations of $(1817,3258,5275)$ and $(4904676969, 10840875082, 15051171563)$.

Answer 2

My paper

"Unsolvable cases of $P^3+Q^3+cR^3=d PQR$", Rocky Mountain J. of Math. (28),No 3, 1998

gives 7 (infinite) generic classes of unsolvability of the original equation. However there are still many (generic) cases to prove! Solutions have been found for all $n$ in the range $0<n\le1000$.

Answer 3

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]

Answer 4

@Haran enquired about case for $n=6$.

Well lately Seiji Tomita has given a numerical solution for it.

He used the equation below to arrive at a numerical solution: $$ (m^2+m+1)^3+(m^2-m+1)^3+(2)^3=(m^2+5)(m^2+m+1)(2)(m^2-m+1). $$

To arrive at $n=6$ we substitute $m=1$. The downside of this is that, when using the above equation one can get only one numerical solution for $n=6$. The equation gives us: $$ 3^3+1^3+2^3=6(3\cdot2\cdot1). $$ Tomita has given a general method through which solutions for different $n$ can be arrived at. His "Computational number theory" web page discusses this in #456. $x^3 + y^3 + z^3 = n x y z$.