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 = 5(u^8-23u^6v^2+60u^4v^4-23u^2v^6+v^8)(u^2+v^2)^2
     y = 3p(-v^2+2uv+u^2)(-v^2-2uv+u^2)(v^8+31u^2v^6-48u^4v^4+31u^6v^2+u^8)
     z = 2(v^4-4u^2v^2+u^4)(v^8-86u^2v^6+42u^4v^4-86u^6v^2+u^8)p
     w = 4p^3(u^2+v^2)(2v^36-1475v^34u^2+104452v^32u^4-3352665v^30u^6+44128586v^28u^8
         -317232160v^26u^10+1400119350v^24u^12-3984500495v^22u^14+7467292666v^20u^16-9211347050v^18u^18
         +7467292666v^16u^20-3984500495v^14u^22+1400119350v^12u^24-317232160v^10u^26+44128586v^8u^28-3352665v^6u^30+104452v^4u^32-1475v^2u^34+2u^36)
          
     p,u,v are arbitrary.
    
 
Proof.

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

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

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

Let (a,b,c,s) =(m^2-4u^2v^2p^6, (2uv)^2, (u^2-v^2)^2, u^2+v^2  ).

Furthermore, let (m,n,r)=( -8u^2v^2p^3(48u^4v^4-25u^2v^6+5v^8-25u^6v^2+5u^8)/(10v^6u^4+5v^2u^8+10v^4u^6+5v^8u^2+v^10+u^10),
                            6(-2u^2v^2+5u^4+5v^4)p^2u^2v^2/(u^6+3v^2u^4+3v^4u^2+v^6),
                            -12u^2v^2p/(u^2+v^2)  ). 

We obtain t = 5/2(u^12-21u^10v^2+15u^8v^4+74u^6v^6+15u^4v^8-21v^10u^2+v^12)/((u^12-90u^10v^2+387u^8v^4-340u^6v^6+387u^4v^8-90v^10u^2+v^12)p).

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