1.Introduction

We show equation ax^3 + by^3 + cz^3 + dw^3 = 0 has the simple parametric solutions.

(2020.10.15): Added: ax^3 + by^3 + (p^3q-a)z^3 - (b+q)w^3 = 0 has a simple parametric solution.


2.Theorem
        
     
     
ax^3 + by^3 + (p^3q-a)z^3 - (b+q)w^3 = 0 has a simple parametric solution.

(x,y,z,w)=(1,p,1,p)
a,b,p,q are arbitrary.
     
     
x^3 + y^3 + 2z^3 + 6a^2w^3 = 0 has the simple parametric solutions.

(x,y,z,w)=(t^3+a, t^3-a, -t^3, -t)

a,t are arbitrary.

There are related identities below.

ax^3 + ay^3 + 2az^3 + 6w^3 = 0
(x,y,z,w)=(t^3+a, t^3-a, -t^3, -at)

4ax^3 + 4ay^3 + az^3 + 3w^3 = 0
(x,y,z,w)=(t^3+a, t^3-a, -2t^3, -2at)

36ax^3 + 36ay^3 + 9az^3 + w^3 = 0
(x,y,z,w)=(t^3+a, t^3-a, -2t^3, -6at)

x^3 + y^3 + 2z^3 + a^2w^3 = 0
(x,y,z,w)=(36t^3+a, 36t^3-a, -36t^3, -6t)

ax^3 + ay^3 + 2az^3 + w^3 = 0
(x,y,z,w)=(36t^3+a, 36t^3-a, -36t^3, -6at)

4x^3 + 4y^3 + z^3 + 4a^2w^3 = 0
(x,y,z,w)=(36t^3+a, 36t^3-a, -72t^3, -6t)



 
Proof.

(u+a)^3 + (u-a)^3  = 2u^3 + 6ua^2.........................................(1)

Substitute u = t^3 to (1), and simplifying (1), we obtain

(u+a)^3 + (u-a)^3 + 2(-t^3)^3 + 6a^2(-t)^3 = 0............................(2)

Let x=u+a, y=u-a, z=-t^3, w=-t, then we obtain a parametric solution as follows.

x^3 + y^3 + 2z^3 + 6a^2w^3=0..............................................(3)
   
(x,y,z,w)=(t^3+a, t^3-a, -t^3, -t)
a,t are arbitrary.

Identity (3) has other expressions below.

ax^3 + ay^3 + 2az^3 + 6w^3=0..............................................(4)
(x,y,z,w)=(t^3+a, t^3-a, -t^3, -at)

4ax^3 + 4ay^3 + az^3 + 3w^3=0.............................................(5)
(x,y,z,w)=(t^3+a, t^3-a, -2t^3, -2at)

36ax^3 + 36ay^3 + 9az^3 + w^3=0...........................................(6)
(x,y,z,w)=(t^3+a, t^3-a, -2t^3, -6at)

x^3 + y^3 + 2z^3 + a^2w^3=0...............................................(7)
(x,y,z,w)=(36t^3+a, 36t^3-a, -36t^3, -6t)

ax^3 + ay^3 + 2az^3 + w^3=0...............................................(8)
(x,y,z,w)=(36t^3+a, 36t^3-a, -36t^3, -6at)

4x^3 + 4y^3 + z^3 + 4a^2w^3=0.............................................(9)
(x,y,z,w)=(36t^3+a, 36t^3-a, -72t^3, -6t)

Q.E.D.