1.Introduction


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

a = -(-16r^4q^4v^2-16u^2p^4q^4-8v^2p^2q^4r^2+8v^2p^4q^2r^2+4v^2p^2q^2r^4+12w^2q^2p^2r^4+12w^2q^2p^4r^2-8r^2q^4u^2p^2+4r^2q^2u^2p^4
  +8r^4q^2u^2p^2+4v^2p^2q^6+4w^2q^6r^2-6w^2q^4p^4-8v^2p^4q^4+4v^2p^6q^2-6w^2q^4r^4+4w^2q^2p^6+4w^2q^2r^6-4w^2p^6r^2-6w^2p^4r^4
  -4w^2p^2r^6+4r^2q^6u^2-8r^4q^4u^2+4r^6q^2u^2+4r^2q^4w^2p^2-w^2q^8-w^2r^8-w^2p^8+4w^2q^6p^2)
  /(r^8+4p^2r^6-8q^2r^6+6p^4r^4-24q^2p^2r^4+30q^4r^4+12q^4r^2p^2+4p^6r^2-24q^2p^4r^2-8q^6r^2+q^8-8q^2p^6-8q^6p^2+30q^4p^4+p^8)
  
b = (8r^4q^4v^2+6u^2p^4q^4+8v^2p^2q^4r^2-4v^2p^4q^2r^2-8v^2p^2q^2r^4-4w^2q^2p^2r^4-8w^2q^2p^4r^2-4r^2q^4u^2p^2-12r^2q^2u^2p^4
  -12r^4q^2u^2p^2+8w^2q^4p^4+16v^2p^4q^4+16w^2q^4r^4-4w^2q^2p^6-4r^2q^6u^2+6r^4q^4u^2-4r^6q^2u^2+8r^2q^4w^2p^2-4r^2q^6v^2
  -4w^2q^6p^2+u^2q^8+u^2p^8+u^2r^8-4u^2q^6p^2-4u^2q^2p^6+4u^2p^6r^2+6u^2p^4r^4+4u^2p^2r^6-4r^6q^2v^2)
  /(r^8+4p^2r^6-8q^2r^6+6p^4r^4-24q^2p^2r^4+30q^4r^4+12q^4r^2p^2+4p^6r^2-24q^2p^4r^2-8q^6r^2+q^8-8q^2p^6-8q^6p^2+30q^4p^4+p^8)  
  
c = -(-6r^4q^4v^2-8u^2p^4q^4+4v^2p^2q^4r^2+12v^2p^4q^2r^2+12v^2p^2q^2r^4+8w^2q^2p^2r^4+4w^2q^2p^4r^2-8r^2q^4u^2p^2+8r^2q^2u^2p^4
  +4r^4q^2u^2p^2+4v^2p^2q^6+4w^2q^6r^2-16w^2q^4p^4-6v^2p^4q^4+4v^2p^6q^2-8w^2q^4r^4+4w^2q^2r^6-16r^4q^4u^2-8r^2q^4w^2p^2+4r^2q^6v^2
  -v^2r^8-v^2q^8-v^2p^8-4v^2p^2r^6-6v^2p^4r^4-4v^2p^6r^2+4u^2q^6p^2+4u^2q^2p^6+4r^6q^2v^2)
  /(r^8+4p^2r^6-8q^2r^6+6p^4r^4-24q^2p^2r^4+30q^4r^4+12q^4r^2p^2+4p^6r^2-24q^2p^4r^2-8q^6r^2+q^8-8q^2p^6-8q^6p^2+30q^4p^4+p^8) 
  
[x,y,z]=[-2pq/(p^2+q^2+r^2), (-q^2+p^2+r^2)/(p^2+q^2+r^2), -2rq/(p^2+q^2+r^2)]  
   
p,q,r,u,v,w are arbitrary.

 
Proof.

ax^2 + by^2 + cz^2 = u^2................................................................(1)

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

cx^2 + ay^2 + bz^2 = w^2................................................................(3)

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

a = (x^4u^2-x^2v^2y^2-x^2z^2w^2-z^2y^2u^2+y^4w^2+z^4v^2)/(z^6-3z^2x^2y^2+y^6+x^6).......(4)

b = (v^2x^4-x^2y^2w^2-x^2z^2u^2+u^2y^4-z^2v^2y^2+z^4w^2)/(z^6-3z^2x^2y^2+y^6+x^6).......(5)

c = (-z^2x^2v^2+u^2z^4+y^4v^2-z^2y^2w^2-y^2x^2u^2+x^4w^2)/(z^6-3z^2x^2y^2+y^6+x^6)......(6)

Let x^2+y^2+z^2=1 and obtain a parametric solution of x^2+y^2+z^2=1 below.

[x,y,z]=[-2pq/(p^2+q^2+r^2), (-q^2+p^2+r^2)/(p^2+q^2+r^2), -2rq/(p^2+q^2+r^2)]..........(7)
p,q,r are 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,q,r]=[3,2,1].
We obtain
[x,y,z]=[6/7, 3/7, 2/7]
[a,b,c]=[-77/127, 315/127, 1540/127].

Let [u,v,w]=[1,2,3], [p,q,r]=[4, 5, 6].
[x,y,z]=[40/77, 27/77, 60/77]
[a,b,c]=[7567/928663, 14379463/928663, -1385748/928663]

In this way, this parametric solution gives infinite solutions.