1.Introduction

Inspired by X^3 + Y^3 + dZ^3 = 0 on MathStackExchange X^3 + Y^3 + dZ^3 = 0, we show a parameter solution of X^3 + Y^3 + dZ^3 =0.


2.Theorem
    
     
    If p^3+q^3+d=0 has a integer solution then,

　　there is a parameter solution of X^3 + Y^3 + dZ^3 = 0 below.
　　
    [X,Y,Z]=[p(2q^3+p^3)(-pb+aq), -q(q^3+2p^3)(-pb+aq), (p-q)(p^2+pq+q^2)(-pb+aq)]

    a,b: arbitrary 
        
Proof.

X^3 + Y^3 + dZ^3 = 0..........................................................(1)

X=pt+a, Y=qt+b, Z=t+c.........................................................(2)

Substitute wquation (2) to equation (1), and simplifying (1),we obtain

(p^3+q^3+d)t^3+(3ap^2+3bq^2+3dc)t^2+(3a^2p+3b^2q+3dc^2)t+a^3+b^3+dc^3=0.......(3)
Substitute d=-p^3-q^3 to equation (3) and  decide c to 3ap^2+3bq^2+3dc=0, then we obtain

c = (ap^2+bq^2)/(p^3+q^3).

t = -1/3(aq^4+2ap^3q+2pbq^3+p^4b)/(qp(p^3+q^3)).

Substitute c and t to equation (2), and obtain a parameter solution.            

   
Q.E.D.　
　　                  
       
3.Examples

Case d=-9,(p,q)=(2,1)

[X,Y,Z]=[20a-40b, -17a+34b, 7a-14b]

Case d=-28,(p,q)=(3,1)

[X,Y,Z]=[87a-261b, -55a+165b, 26a-78b]