1.Introduction

I show the parametric solutions for x^2 + y^2 = z^n + w^n and x^3 + y^3 = z^n + w^n.



2.Theorem
      
 


There is a parametric solution of x^2 + y^2 = z^n + w^n,

      x = (1+p^2)^(1/2*m-1)*(-2*a^(1/2*m)*p^2-2*p*b^(1/2*m)+(1+p^2)*a^(1/2*m))

      y = (1+p^2)^(1/2*m-1)*(-2*a^(1/2*m)*p-2*b^(1/2*m)+(1+p^2)*b^(1/2*m))

      z = a^(m/n)*(1+p^2)^(m/n)

      w = b^(m/n)*(1+p^2)^(m/n).

 condition: 
    
     m=lcm(2,n)
     a, b, p, n: arbitrary


There is a parametric solution of x^3 + y^3 = z^n + w^n,

      x = (a^m-b^m)^(1/3*m-1)*a^(1/3*m)*(2*b^m+a^m)

      y = (a^m-b^m)^(1/3*m-1)*b^(1/3*m)*(-2*a^m-b^m)

      z = a^(m/n)*(a^m-b^m)^(m/n)

      w = b^(m/n)*(a^m-b^m)^(m/n).

 condition: 
    
     m=lcm(3,n)
     a, b, n: arbitrary

     
Proof.

x^2 + y^2 = z^n + w^n......................................................(1)
Let m = lcm(2,n).
Set x=pt+a^(m/2), y=t+b^(m/2), z=a^(m/n), w=b^(m/n)........................(2)

Then we obtain t = -2(a^(1/2m)p+b^(1/2m))/(1+p^2).
Substitute t to (2), and we obtain a solution.


x^3 + y^3 = z^n + w^n......................................................(3)
Let m = lcm(3,n).
Set x=pt+a^(m/3), y=t+b^(m/3), z=a^(m/n), w=b^(m/n)........................(4)

Then we obtain p = -(b/a)^(2/3m), t = -3b^(1/3m)(a^(1/3m))^3/((a^(1/3m))^3-(b^(1/3m))^3).
Substitute p and t to (4), and we obtain a solution.


 
Q.E.D.
　



3.Example



Case. n=3

( -(1+p^2)^2*(p^2*a^3+2*p*b^3-a^3) )^2 + ( (1+p^2)^2*(b^3*p^2-2*a^3*p-b^3) )^2 = ( a^2*(1+p^2)^2 )^3 +( b^2*(1+p^2)^2 )^3

( -a*(2*b^3+a^3) )^3 + ( b*(2*a^3+b^3) )^3 = ( a(b^3-a^3) )^3 + ( b(b^3-a^3) )^3 (same as Vieta's identity)

Case. n=4

( -(1+p^2)*(p^2*a^2+2*p*b^2-a^2) )^2 +( (1+p^2)*(b^2*p^2-2*a^2*p-b^2) )^2 = ( (1+p^2)*a )^4 + ( (1+p^2)*b )^4

( -(-a^12+b^12)^3*a^4*(2*b^12+a^12) )^3 + ( (-a^12+b^12)^3*b^4*(b^12+2*a^12) )^3 = ( a^3*(-a^12+b^12)^3 )^4 + ( b^3*(-a^12+b^12)^3 )^4