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

Oliver Couto found a simple parametric solution of  ax^6 + by^6 + cz^6 = w^2 as follows.


2.Theorem
        
     
     ax^6 + by^6 + cz^6 = w^2  has a parametric solution.
     
     x = m^4+6m^2n^2+n^4
     y = 2m^4+2n^4
     z = (-n^2-2mn+m^2)(-n^2+2mn+m^2)
     w = 18mn(-n+m)(n+m)(m^2+n^2)(n^4-2n^3m+2m^2n^2+2m^3n+m^4)(n^4+2n^3m+2m^2n^2-2m^3n+m^4)
     
     m,n are arbitrary.
     c=-(a+b).
     
 
Proof.

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

Substitute x=a+2b, y=2a+b, z=a-b, c=-(a+b) to equation (1) of LHS, then we obtain an identity

a(a+2b)^6+b(2a+b)^6-(a+b)(a-b)^6 = 81ab(a+b)(b^2+ab+a^2)^2.....................(2)

Furthermore, let (a,b) =( (m^2-n^2)^2 , (2mn)^2  ).

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



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