1.Introduction

Previously, we studied the parametric solutions of ax^6 + by^6 + cz^6 = w^2 below.
ax^6 + by^6 + cz^6 = w^2 Ⅰ
ax^6 + by^6 + cz^6 = w^2 Ⅱ

We show the other parametric solution of  ax^6 + by^6 + cz^6 = w^2.


2.Theorem
        
     
     ax^6 + by^6 + cz^6 = w^2  has a parametric solution.
     
     x = p(-3b+4s^2)(s^4+63s^2b-108b^2)
     y = p(3b+s^2)(44s^4-153s^2b+108b^2)
     z = 5s^2(4s^4-27s^2b+27b^2)
     w = (s^2-2b)(64s^16+702000s^14b-9744516s^12b^2+81602073s^10b^3-338831181s^8b^4+747501291s^6b^5-898942293s^4b^6+556950168s^2b^7-139237542b^8)p^3s
     
     b,p,s are arbitrary.
    
 
Proof.

ax^6 + by^6 + cz^6 = w^2.......................................................(1)

Substitute x=pt+1, y=pt-1, z=t, w=mt^3+nt^2+rt+s to equation (1), then we obtain

(bp^6+c+ap^6-m^2)t^6+(6ap^5-6bp^5-2nm)t^5+(-2rm-n^2+15ap^4+15bp^4)t^4+(-20bp^3-2sm-2rn+20ap^3)t^3
+(15bp^2-2sn-r^2+15ap^2)t^2+(6ap-6bp-2sr)t+a-s^2+b.............................(2)

Let (a,c) =( s^2-b , -p^6s^2+m^2  ).

Furthermore, let (m,n,r)=( p^3(-56s^4b+s^6+162s^2b^2-108b^3)/(s^5),
                           3p^2(s^4+6s^2b-6b^2)/(s^3),
                           3p(s^2-2b)/s ). 

We obtain t = -5/6s^2(4s^4-27s^2b+27b^2)/((4s^6+19s^4b-81s^2b^2+54b^3)p).

Finally, we can obtain a parametric solution for equation (1).
     
Q.E.D.