1.Introduction


We show simultaneous equation {ax^5 + by^5 + cz^5=u^5, bx^5 + cy^5 + az^5=v^5, cx^5 + ay^5 + bz^5=w^5} has a parametric solution.
     
2.Theorem
     
A simultaneous equation {ax^5 + by^5 + cz^5=u^5, bx^5 + cy^5 + az^5=v^5, cx^5 + ay^5 + bz^5=w^5} has a parametric solution.

a = 500(46539306640625q^45+18463134765625p^5q^40+219726562500p^10q^35-280273437500p^15q^30+31074218750p^20q^25-649531250p^25q^20+10312500p^30q^15-47500p^35q^10+25p^40q^5+p^45)q^5
  /(286102294921875q^50+12969970703125p^10q^40-285644531250p^20q^30+8910156250p^30q^20-140625p^40q^10+p^50)

b = -(46253204345703125q^50+206756591796875p^10q^40+31359863281250p^20q^30+1402343750p^30q^20+165625p^40q^10-p^50)
  /(286102294921875q^50+12969970703125p^10q^40-285644531250p^20q^30+8910156250p^30q^20-140625p^40q^10+p^50)

c = 500(46539306640625q^45-18463134765625p^5q^40+219726562500p^10q^35+280273437500p^15q^30+31074218750p^20q^25+649531250p^25q^20+10312500p^30q^15+47500p^35q^10+25p^40q^5-p^45)q^5
  /(286102294921875q^50+12969970703125p^10q^40-285644531250p^20q^30+8910156250p^30q^20-140625p^40q^10+p^50)

  
[x,y,z,u,v,w]=[p^5+25q^5, p^5-25q^5, 10p^3q^2, p^5+75q^5, p^5-75q^5, -50pq^4] 
   
p,q are arbitrary.

 
Proof.

ax^5 + by^5 + cz^5...........................................................................(1)

bx^5 + cy^5 + az^5...........................................................................(2)

cx^5 + ay^5 + bz^5...........................................................................(3)

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

a = (-y^5v^5x^5+y^10w^5-y^5z^5u^5-z^5w^5x^5+z^10v^5+u^5x^10)/(-3x^5y^5z^5+x^15+z^15+y^15)....(4)

b = (v^5x^10-x^5y^5w^5-x^5z^5u^5+z^10w^5+y^10u^5-y^5z^5v^5)/(-3x^5y^5z^5+x^15+z^15+y^15).....(5)

c = (-x^5y^5u^5+w^5x^10-x^5z^5v^5-z^5y^5w^5+z^10u^5+y^10v^5)/(-3x^5y^5z^5+x^15+z^15+y^15)....(6)

Let x^5+y^5+z^5=u^5+v^5+w^5 and obtain one of parametric solutions of x^5+y^5+z^5=u^5+v^5+w^5 below.
See A collection of the parametric solutions for equal sums of fifth powers  

[x,y,z,u,v,w]=[p^5+25q^5, p^5-25q^5, 10p^3q^2, p^5+75q^5, p^5-75q^5, -50pq^4]................(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]=[801, -799, 40]
[u,v,w]=[2401, -2399, -800]
[a,b,c]=[26524223404926160239861772816000/322136807526900862975856000001, -52076705828949198270464169599999/322136807526900862975856000001, 25874619231549938893578252784000/322136807526900862975856000001].