1.Introduction


We show a parametric solution of ax1^n + bx2^n + cx3^n = ay1^n + by2^n + cy3^n, n=2,6.


2.Method
        

ax1^n+bx2^n+cx3^n = ay1^n+by2^n+cy3^n..........................................(1)

Let x1=pt+1, x2=qt+1, x3=t+1, y1=pt-1, y2=qt-1, y3=t-1.........................(2)

Set n= 2.
Substitute (2) to (1) and simplifying (1), we obtain
(4ap+4bq+4c)t=0................................................................(3)

Set n= 6, we obtain 
(12c+12ap^5+12bq^5)t^5+(40c+40bq^3+40ap^3)t^3+(12c+12bq+12ap)t=0...............(4)

Equating to zero the coefficient of t in (4), then we obtain
c = -ap-bqa6.

To obtain the rational solution t of (4), the discriminant must be square as follows. 

v^2 = (480p^6-480p^2+480p^4-480p^8)a^2
    +(-480q^5p^3-480p^5q^3+480p^5q+480qp^3+480pq^3-960pq+480q^5p)ba
    +(480q^6+480q^4-480q^2-480q^8)b^2..........................................(5)
    
By using a known solution (a,b,v)=(q-q^3,p^3-p,0), a parametric solution is given by  

a = -q*(q-1)*(q+1)*(40*q^8-40*q^6-80*q^5*p+80*q^5*p^3-40*q^4+40*q^2*p^2+40*q^2-80*q^2*p^4
  +40*q^2*p^6+80*p*q-80*q*p^3+3+40*p^2-80*p^4+40*p^6)
  
b = p*(p-1)*(p+1)*(40*p^8-40*p^6+80*p^5*q^3-80*p^5*q-40*p^4+40*p^2+40*p^2*q^6+40*q^2*p^2
  -80*p^2*q^4+80*p*q-80*p*q^3+3+40*q^6+40*q^2-80*q^4)
  
c = -p*q*(p+q)*(p-q)*(40*p^8-80*p^6+40*q^2*p^6+80*p^5*q^3-80*p^5*q+40*p^4-80*q^2*p^4
  +80*q*p^3-160*q^3*p^3+80*q^5*p^3+40*p^2*q^6-80*p^2*q^4+80*q^2*p^2-80*q^5*p+80*p*q^3
  +3+40*q^4+40*q^8-80*q^6)
        
x1 = -20*p*q^3+20*p^2-20*p^4+20*p*q+3　　                  
x2 = -20*q^4+20*p*q-20*q*p^3+20*q^2+3
x3 = -20*q^3+20*p-20*p^3+20*q+3

y1 = -20*p*q^3+20*p^2-20*p^4+20*p*q-3
y2 = -20*q^4+20*p*q-20*q*p^3+20*q^2-3
y3 = -20*q^3+20*p-20*p^3+20*q-3.

p and q are arbitrary.
　　                  
       
3.Example


Case: (p, q)=(3, 2)

(a, b, c)=(-60001, 480004, -780005)
(x1,x2,x3)=(599, 399, 199)
(y1,y2,y3)=(601, 401, 201)

-60001*599^2 + 480004*399^2 - 780005*199^2 = -60001*601^2 + 480004*401^2 - 780005*201^2

-60001*599^6 + 480004*399^6 - 780005*199^6 = -60001*601^6 + 480004*401^6 - 780005*201^6