1.Introduction

We show if n is even and n > 3 then X1^3 + X2^3 +...+ Xn^3=0 has infinitely many integer solutions.
     
     
2.Theorem
     
When n is even and n > 3, then X1^3 + X2^3 +...+ Xn^3=0 has infinitely many integer solutions.
 
 
Proof.

X1^3 + X2^3 +...+ Xn^3=0..........................................(1)

First, we obtain a  parametric solution for n=4.

X1^3 + X2^3 + X3^3 + X4^3=0.......................................(2)

Let X1=p1t+a, X2=p2t-a, X3=p3t+b, X4=p4t-b,(p1,p2,p3,p4)=(3,4,5,-6) then we obtain a parametric solution. 

(X1,X2,X3,X4)=(28*a^2-3*b^2+11*a*b, 21*a^2-4*b^2-11*a*b, 35*a^2+6*b^2+7*a*b, -42*a^2-5*b^2-7*a*b)

Next, we obtain a parametric solution for n=6.

X1^3+X2^3+X3^3+X4^3+X5^3+X6^3=0

Let X5=p5t+c, X6=p5t-c, we obtain a parametric solution.

(X1,X2,X3,X4,X5,X6)=(-252*c^6*a-396*c^6*b+343*a^7-147*a^5*b^2+21*a^3*b^4-a*b^6,
                     -336*c^6*a-528*c^6*b-343*a^7+147*a^5*b^2-21*a^3*b^4+a*b^6,
                     -420*c^6*a-660*c^6*b+343*b*a^6-147*a^4*b^3+21*a^2*b^5-b^7,
                      504*c^6*a+792*c^6*b-343*b*a^6+147*a^4*b^3-21*a^2*b^5+b^7,
                      (7*a^2-b^2)*(49*a^4-14*b^2*a^2+42*c^3*a+66*c^3*b+b^4)*c,
                      (7*a^2-b^2)*(-49*a^4+14*b^2*a^2+42*c^3*a+66*c^3*b-b^4)*c)

And furthermore, we obtain a parametric solution for n=8.

X1^3+X2^3+X3^3+X4^3+X5^3+X6^3+X7^3+X8^3=0

Let X7=p5t+d, X8=p5t-d.

(X1,X2,X3,X4,X5,X6,X7,X8)=(-126*a*d^6-378*a*d^4*c^2-378*a*d^2*c^4-126*a*c^6-198*b*d^6-594*b*d^4*c^2-594*b*d^2*c^4-198*b*c^6+343*a^7-147*a^5*b^2+21*a^3*b^4-a*b^6,
                           -168*a*d^6-504*a*d^4*c^2-504*a*d^2*c^4-168*a*c^6-264*b*d^6-792*b*d^4*c^2-792*b*d^2*c^4-264*b*c^6-343*a^7+147*a^5*b^2-21*a^3*b^4+a*b^6,
                           -210*a*d^6-630*a*d^4*c^2-630*a*d^2*c^4-210*a*c^6-330*b*d^6-990*b*d^4*c^2-990*b*d^2*c^4-330*b*c^6+343*b*a^6-147*a^4*b^3+21*a^2*b^5-b^7,
                            252*a*d^6+756*a*d^4*c^2+756*a*d^2*c^4+252*a*c^6+396*b*d^6+1188*b*d^4*c^2+1188*b*d^2*c^4+396*b*c^6-343*b*a^6+147*a^4*b^3-21*a^2*b^5+b^7,
                            (7*a^2-b^2)*(49*c*a^4-14*c*b^2*a^2+21*a*c^4+21*a*d^4+42*a*d^2*c^2+66*c^2*b*d^2+33*b*d^4+c*b^4+33*c^4*b),
                           -(7*a^2-b^2)*(49*c*a^4-14*c*b^2*a^2-21*a*c^4-21*a*d^4-42*a*d^2*c^2-66*c^2*b*d^2-33*b*d^4+c*b^4-33*c^4*b),
                            (7*a^2-b^2)*(49*d*a^4-14*d*b^2*a^2+21*a*c^4+21*a*d^4+42*a*d^2*c^2+33*c^4*b+d*b^4+33*b*d^4+66*c^2*b*d^2),
                           -(7*a^2-b^2)*(49*d*a^4-14*d*b^2*a^2-21*a*c^4-42*a*d^2*c^2-21*a*d^4-33*c^4*b+d*b^4-33*b*d^4-66*c^2*b*d^2))


Similarly, we can obtain a parametric solution of (1) if n is even and n > 3.

Thus equation (1) has infinitely many integer solutions if n is even and n > 3.

Q.E.D.