1.Introduction


We show simultaneous equation {ax^3 + by^3 + cz^3=u^3,bx^3 + cy^3 + az^3=v^3,cx^3 + ay^3 + bz^3=w^3} has a parametric solution.
     
2.Theorem
     
A simultaneous equation {ax^3 + by^3 + cz^3=u^3,bx^3 + cy^3 + az^3=v^3,cx^3 + ay^3 + bz^3=w^3} has a parametric solution.

a = (97686p^12v^3-54u^3p^3+u^3+531441p^24w^3+1215u^3p^6-14580u^3p^9+98415u^3p^12-373977u^3p^15+708588u^3p^18-27w^3p^3
  +972w^3p^6-13851w^3p^9+99144w^3p^12-373977w^3p^15+708588w^3p^18-531441w^3p^21-373977p^15v^3+708588p^18v^3-531441p^21v^3
  -531441u^3p^21+531441u^3p^24+729v^3p^6-13122v^3p^9+531441v^3p^24)
  /(1-41553p^9-81p^3+2916p^6+295245p^12-1121931p^15+2125764p^18-1594323p^21+1594323p^24)
  
b = (98415p^12v^3-27u^3p^3+v^3-54v^3p^3+531441p^24w^3+972u^3p^6-13851u^3p^9+99144u^3p^12-373977u^3p^15+708588u^3p^18
  +729w^3p^6-13122w^3p^9+97686w^3p^12-373977w^3p^15+708588w^3p^18-531441w^3p^21-373977p^15v^3+708588p^18v^3-531441p^21v^3
  -531441u^3p^21+531441u^3p^24+1215v^3p^6-14580v^3p^9+531441v^3p^24)
  /(1-41553p^9-81p^3+2916p^6+295245p^12-1121931p^15+2125764p^18-1594323p^21+1594323p^24)
  
c = (99144p^12v^3+w^3-27v^3p^3+531441p^24w^3+729u^3p^6-13122u^3p^9+97686u^3p^12-373977u^3p^15+708588u^3p^18-54w^3p^3
  +1215w^3p^6-14580w^3p^9+98415w^3p^12-373977w^3p^15+708588w^3p^18-531441w^3p^21-373977p^15v^3+708588p^18v^3-531441p^21v^3
  -531441u^3p^21+531441u^3p^24+972v^3p^6-13851v^3p^9+531441v^3p^24)
  /(1-41553p^9-81p^3+2916p^6+295245p^12-1121931p^15+2125764p^18-1594323p^21+1594323p^24)
  
[x,y,z]=[1-9p^3, 9p^4, -3p(-1+3p^3)]  
   
p,u,v,w are arbitrary.

 
Proof.

ax^3 + by^3 + cz^3=u^3.................................................................(1)

bx^3 + cy^3 + az^3=v^3.................................................................(2)

cx^3 + ay^3 + bz^3=w^3.................................................................(3)

We obtain the solution of equation (1),(2), and (3) for {a,b,c} below.

a = (x^6u^3-x^3z^3w^3-x^3y^3v^3+y^6w^3-z^3y^3u^3+v^3z^6)/(-3z^3x^3y^3+y^9+z^9+x^9).....(4)

b = (v^3x^6-x^3y^3w^3-x^3z^3u^3+z^6w^3-y^3z^3v^3+y^6u^3)/(-3z^3x^3y^3+y^9+z^9+x^9).....(5)

c = (-z^3y^3w^3+u^3z^6+x^6w^3-z^3x^3v^3+y^6v^3-x^3y^3u^3)/(-3z^3x^3y^3+y^9+z^9+x^9)....(6)

Let x^3+y^3+z^3=1 and obtain a parametric solution of x^3+y^3+z^3=1 below.

[x,y,z]=[1-9p^3, 9p^4, -3p(-1+3p^3)]...................................................(7)
p is arbitrary.

Substitute the parametric solution (7) to equation (4),(5), and (6).

Hence we obtain a parametric solution of simultaneous equation.


Q.E.D.


3.Example

Let [u,v,w]=[1,2,3], [p]=[3].
We obtain
[x,y,z]=[-242, 729, -720]
[a,b,c]=[5212972163368163467/434414346325663897, 5212972146104311337/434414346325663897, 5212972158251425488/434414346325663897]

Let [u,v,w]=[1,2,3], [p]=[4].
[x,y,z]=[-575, 2304, -2292]
[a,b,c]=[5302740051288365198401/441895004254554434497, 5302740050740269940040/441895004254554434497, 5302740051135324503451/441895004254554434497]

In this way, this parametric solution gives infinite solutions.