1.Introduction

It's showed that X^3+Y^3+Z^3-2XYZ = 19 has infinitely many integral solutions on MathStackExchange website[1].

We show that X^3+Y^3+Z^3-nXYZ = (n-3)(n^2+3n+9) has infinitely many integral solutions for arbitrary n.


2.Theorem
        
     

    X^3+Y^3+Z^3-nXYZ = (n-3)(n^2+3n+9) has infinitely many integral solutions as follows.
    
    (m)^3+(n-m)^3+(-3)^3-n(m)(n-m)(-3) = (n-3)(n^2+3n+9)
    
    m,n are arbitrary.           

 
Proof.

X^3+Y^3+Z^3-nXYZ....................................................(1)

Put X=t+a, Y=-t+b, Z=c..............................................(2)

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

(nc+3a+3b)t^2+(-(-na+nb)c+3a^2-3b^2)t+a^3-nabc+b^3+c^3=0.............(3)

Equating to zero the coefficient of t and t^2, then we obtain b = -1/3nc-a.

Substitute b = -1/3nc-a to equation (1) and let c=-3 and m=t+a, then we obtain equation (4).

X^3+Y^3+Z^3-nXYZ = (n-3)(n^2+3n+9)..................................(4)
(X,Y,Z) = (m, n-m, -3).
m,n are arbitrary.

Hence equation (4) has infinitely many integral solutions.
   
Q.E.D.　
 
　　                  
       
3.Examples



X^3+Y^3+Z^3-XYZ  = ±26 has infinitely many integral solutions.

X^3+Y^3+Z^3-2XYZ = ±19              "

X^3+Y^3+Z^3-3XYZ = 0                 "

X^3+Y^3+Z^3-4XYZ = ±37              "

 

4.Reference

[1]: StackExchange: For what N does N=x^3+y^3+z^3-2xyz have infinite integer solutions?