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In

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?

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    $\begingroup$ This curve has a rational point at $(x:y:z)=(1:2:3)$ (and infinitely many others, such as $(19:21:-52)$ ). If I did this right it's isomorphic with $Y^2 + Y = X^3 - 54 X - 88$, which has torsion ${\bf Z} / 3{\bf Z}$ (as usual in this family) and rank 1, see e.g. <lmfdb.org/EllipticCurve/Q/189/b/2>. So yes, one can start with $(1:2:3)$ and its permutations and generate all rational solutions with "chords and tangents". $\endgroup$
    – Noam D. Elkies
    Commented Feb 21, 2021 at 18:57
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    $\begingroup$ But rational solutions to the Weierstrass equation will always correspond to integer solutions of the homogeneous equation in OP’s question, @dodd . $\endgroup$
    – Lubin
    Commented Feb 21, 2021 at 19:34
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    $\begingroup$ @dodd as usual, each rational point on the elliptic curve corresponds to a unique primitive solution (up to multiplying $(x,y,z)$ by $-1$). $$ $$ Haran: 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 $(4904676969, 10840875082, 15051171563)$. $\endgroup$
    – Noam D. Elkies
    Commented Feb 21, 2021 at 20:07
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    $\begingroup$ @NoamD.Elkies and even more characters can be used for human-readable information that might be of use to someone down the track. :-) $\endgroup$
    – David Roberts ♦
    Commented Feb 22, 2021 at 2:57
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    $\begingroup$ The method to transform an arbitrary smooth cubic into a Weierstrass form is standard and is explained for instance in the lovely book by Silverman-Tate, see section 1.3 or appendix B there. It is also implemented in Sage as EllipticCurve_from_cubic $\endgroup$
    – Wojowu
    Commented Feb 22, 2021 at 12:03

4 Answers 4

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(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 $(4904676969, 10840875082, 15051171563)$.

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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 $0<n\le1000$.

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  • $\begingroup$ "For all n", what do you mean? All d, certain c? $\endgroup$
    – Wolfgang
    Commented Apr 1, 2021 at 15:20
  • $\begingroup$ Sorry I was not clear, by 'original equation' I meant x^3+y^3+z^3=nxyz..Also, I should have written *Solutions have been found for all solvable n-cases <=1000. $\endgroup$
    – Erik Dofs
    Commented Apr 1, 2021 at 19:50
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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]
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  • $\begingroup$ Does your table give, for each $n$, the smallest solution to (1)? The solution you give for $n=177$ is quite large! $\endgroup$
    – Gerry Myerson
    Commented Apr 7, 2021 at 0:32
  • $\begingroup$ Since solutions are obtained by generators, I guess solutions are smallest for rank=1. However, it is not always the smallest solution for rank>1. $\endgroup$
    – Tomita
    Commented Apr 7, 2021 at 2:55
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@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. $x^3 + y^3 + z^3 = n x y z$.

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