x^3+y^3+z^3 = nxyz

Inspired by mathoverflow's problem, I decided to find the genenal solution of x^3+y^3+z^3 = nxyz.
x^3+y^3+z^3=6xyz

We show diophantine equation x^3+y^3+z^3=nxyz has infinitely many positive integer solutions.
Furthermore, we show the nemerical solutions with n<500.

Erik Dofs found a solution for n=429 below.(2021.3.29)

x = 260368915802354995442404638422249663506509113438271385614244973237991273982662095
y = 94191416816891897810689525972918108983946101211470784286772974217504230998367383
z = 1757111002107224787045574271833566926442724424835235300437893329702042769905782

Erik Dofs found a solution for n=442 below.(2021.3.29)

x = 91981091293399054566820692644530476835209719559605599712883693763224039274564252983294703523377175766937704906138942308334588052436844816617714107906618796522694473773012949863479785
y = 2993246349314577720671836534886868498291119906905805764557361706933991826218041244325271253053045596584995759683968270082224927156864639612601815045003330032014578721206763700858214
z = 6397251542698242492781944590787745944564279487881453400043005304419518118431253510798456960196085105543736679328362681550974444472610104242364936341586464234597549805377140038442171


x^3+y^3+z^3 = nxyz.........................................................................(1)

Let s=x+y+z, equation (1) reduces to

V^2 = -3U^4+(12+2n)U^3+(-18-6n+n^2)U^2+(6n-2n^2)U+9+n^2-6n.................................(2)

This quartic equation is birationally equivalent to an 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.

U = (-12n^2+2nX-6X+108)/Y
V = (-5832+60n^3X+324n^2X-972nX-8n^3Y+72n^4+3888n-1620X+216Y-3X^3+162X^2-432n^3-18n^2X^2+X^3n)/(Y^2)
X = (-6V+18-12n+2nV+2n^2+6Un-2Un^2)/(U^2)
Y = (-4Un^3+4n^3-12U^2n^2+4n^2V+24Un^2-36n^2-36Un-24nV+108n+108U^2-108+36V)/(U^3)

Let n=m^2+5, we know equation (2) has a rational solution (U,V)=( m^2+2,4m(m^2+2) ).

Then we obtain a parametric solution of equation (1) below.

(x,y,z)=(2, m^2+1-m, m^2+1+m).
m is arbitrary.
We know x,y,z are all positive.

The corresponding point is P(X,Y)=( 1/4(m^8+26m^6+27m^4-22m^2+49)/m^2, 1/4(m^12+17m^10+106m^8+257m^6+186m^4-287m^2-343)/m^3 ).

Using 2P(X,Y) we obtain other parametric solution of equation (1) below.

(x,y,z)=( 2(m^12+39m^10+174m^8+259m^6+78m^4-57m^2+49)(m^2+m+1)(m^2-m+1)
         ,(m^12+2m^11+3m^10+8m^9+4m^8-10m^7+19m^6-56m^5+56m^4-30m^3+55m^2+20m+7)(m^6+m^5+4m^4-3m^3+12m^2+m+7)
         ,(m^6-m^5+4m^4+3m^3+12m^2-m+7)(m^12-2m^11+3m^10-8m^9+4m^8+10m^7+19m^6+56m^5+56m^4+30m^3+55m^2-20m+7)
        ).
We know x,y,z are all positive.

This point P is of infinite order, and the multiples kP, k = 2, 3, ...give infinitely many points.

This quartic equation (2) has infinitely many parametric solutions.

Hence we can obtain infinitely many positive integer solutions for equation (1).


About condition of positive solution.
At least x/s = U = (-12n^2+2nX-6X+108)/Y must be pisitive.

Case 1: Y>0 and X>(6n^2-54)/(n-3).
Few solutions are relevant to this case.

Case 2: Y<0 and X<(6n^2-54)/(n-3).
Almost positive solutions are relevant to this case.




Numerical solutions.
Only smallest(maybe) solutions are shown.
n<500
[ n  rank x y z]

[1][0]
[2][0]
[3][0][1, 1, 1]
[4][0]
[5][0][1, 2, 1]
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