1.Introduction


We show simultaneous equation {ax^4 + by^4 + cz^4=u^4, bx^4 + cy^4 + az^4=v^4, cx^4 + ay^4 + bz^4=w^4} has a parametric solution.
     
2.Theorem
     
A simultaneous equation {ax^4 + by^4 + cz^4=u^4, bx^4 + cy^4 + az^4=v^4, cx^4 + ay^4 + bz^4=w^4} has a parametric solution.

a = 48/7(25p^6-140p^5q+1149p^4q^2+1400p^3q^3-1173p^2q^4-1260pq^5+1599q^6)
  / (775p^6-2880p^5q+111p^4q^2+28800p^3q^3+34653p^2q^4-25920pq^5+14061q^6)

b = 80/7(65p^6-84p^5q-507p^4q^2+840p^3q^3+3939p^2q^4-756pq^5+663q^6)
  / (775p^6-2880p^5q+111p^4q^2+28800p^3q^3+34653p^2q^4-25920pq^5+14061q^6)

c = -15/7(65p^6+448p^5q+921p^4q^2-4480p^3q^3+1083p^2q^4+4032pq^5+2091q^6)
  / (775p^6-2880p^5q+111p^4q^2+28800p^3q^3+34653p^2q^4-25920pq^5+14061q^6)
  
[x,y,z,u,v,w]=[p-5q, 2p+4q, 3p-q, p+5q, 2p-4q, 3p+q] 
   
p,q are arbitrary.

 
Proof.

ax^4 + by^4 + cz^4........................................................................(1)

bx^4 + cy^4 + az^4........................................................................(2)

cx^4 + ay^4 + bz^4........................................................................(3)

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

a = (-y^4v^4x^4-y^4z^4u^4+y^8w^4+z^8v^4-z^4w^4x^4+u^4x^8)/(-3x^4y^4z^4+z^12+x^12+y^12)....(4)

b = (v^4x^8-x^4z^4u^4-x^4y^4w^4+w^4z^8-y^4v^4z^4+y^8u^4)/(-3x^4y^4z^4+z^12+x^12+y^12).....(5)

c = (-x^4z^4v^4+w^4x^8+y^8v^4-x^4y^4u^4+z^8u^4-z^4y^4w^4)/(-3x^4y^4z^4+z^12+x^12+y^12)....(6)

Let x^4+y^4+z^4=u^4+v^4+w^4 and obtain one of parametric solutions of x^4+y^4+z^4=u^4+v^4+w^4 below.
See A^4+ B^4+ C^4 = D^4+ E^4+ F^4  PART 2  

[x,y,z,u,v,w]=[p-5q, 2p+4q, 3p-q, p+5q, 2p-4q, 3p+q]......................................(7)
p,q 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 [p,q]=[1,2].
We obtain
[x,y,z]=[9, 10, 1]
[u,v,w]=[11, 6, 5]
[a,b,c]=[2821872/5955397, 6868240/5955397, -3734715/5955397].







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