1.Introduction

I showed the parametric solution for x^3+y^3+z^3 = t^n in the past[1].

I obtained a different solution from it.

[1].Solutions  of  x3 + y3 + z3 = tn



2.Theorem
      
 a,b,p,n,m:integer

 condition: 
    
     n>=2.
     m=lcm(3,n)
     

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

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

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

      z = -b( b^3-a^(3n) )^(m/3)

      t = a^3( b^3-a^(3n) )^(m/n)


     
Proof.

(x+a^n)^3+(px+b)^3+(-b)^3-(a^3)^n = (1+p^3)x^3+(3a^n+3bp^2)x^2+(3(a^n)^2+3b^2p)x...............(1)

Set p=-(a^n)^2/(b^2) and x= -3a^nb^3/(b^3-(a^n)^3), then we obtain above identity.



 
Q.E.D.
　

 


3.Example

I show only the case of n=2, 3, 4, and 5.


Case. n=2

      {a^2(a^6-b^3)(2b^3+a^6)}^3 + {-b(a^6-b^3)(2a^6+b^3)}^3 + {-b(a^6-b^3)^2}^3
      = {a^3(a^6-b^3)^3}^2

Case. n=3

      {a^3(2b^3+a^9)}^3 + {-b(2a^9+b^3)}^3 + {-b(a^9-b^3)}^3
      = {a^3(a^9-b^3)}^3

Case. n=4

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

Case. n=5

      {-a^5(b^3-a^15)^4(2b^3+a^15)}^3 + {b(b^3-a^15)^4(2a^15+b^3)}^3 + {-b(b^3-a^15)^5}^3
      = {a^3(b^3-a^15)^3}^5