1.Introduction

We show if n is odd and n >= 5 then X1^3 + X2^3 +...+ Xn^3=0 has infinitely many integer solutions.
     
     
2.Theorem
     
When n is odd and n >= 5, 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=5.

X1^3 + X2^3 + X3^3 + X4^3+ X5^3=0.......................................(2)

Let X1=pt+a, X2=t-a, X3=t+b, X4=t-b, X5=t then we obtain a parametric solution. 

(X1,X2,X3,X4,X5)=(a*(6*a^4*b^2+24*a^2*b^4+16*b^6+3*a^6),
                 -a*(6*a^4*b^2+3*a^6-8*b^6),
                  b*(-12*a^5*b-12*a^3*b^3+3*a^6-6*a^4*b^2-12*a^2*b^4-8*b^6),
                 -b*(12*a^5*b+12*a^3*b^3+3*a^6-6*a^4*b^2-12*a^2*b^4-8*b^6),
                 -12*b^2*(a^2+b^2)*a^3)

Next, we obtain a parametric solution for n=7.

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

Let X5=t+c, X6=t-c, X7=t then we obtain a parametric solution.

(X1,X2,X3,X4,X5,X6,X7)=(a*(6*a^4*b^2+6*a^4*c^2+24*a^2*b^4+48*a^2*b^2*c^2+24*a^2*c^4+16*b^6+48*b^4*c^2+48*b^2*c^4+16*c^6+5*a^6),
                       -a*(6*a^4*b^2+6*a^4*c^2+5*a^6-8*b^6-24*b^4*c^2-24*b^2*c^4-8*c^6),
                       -12*a^5*b^2-12*a^5*c^2-12*a^3*b^4-24*a^3*b^2*c^2-12*a^3*c^4+5*b*a^6-6*a^4*b^3-6*b*a^4*c^2-12*a^2*b^5-24*a^2*b^3*c^2-12*b*a^2*c^4-8*b^7-24*b^5*c^2-24*b^3*c^4-8*b*c^6,
                       -12*a^5*b^2-12*a^5*c^2-12*a^3*b^4-24*a^3*b^2*c^2-12*a^3*c^4-5*b*a^6+6*a^4*b^3+6*b*a^4*c^2+12*a^2*b^5+24*a^2*b^3*c^2+12*b*a^2*c^4+8*b^7+24*b^5*c^2+24*b^3*c^4+8*b*c^6,
                       -12*a^5*b^2-12*a^5*c^2-12*a^3*b^4-24*a^3*b^2*c^2-12*a^3*c^4+5*c*a^6-6*c*a^4*b^2-6*a^4*c^3-12*c*a^2*b^4-24*a^2*b^2*c^3-12*a^2*c^5-8*c*b^6-24*b^4*c^3-24*b^2*c^5-8*c^7,
                       -12*a^5*b^2-12*a^5*c^2-12*a^3*b^4-24*a^3*b^2*c^2-12*a^3*c^4-5*c*a^6+6*c*a^4*b^2+6*a^4*c^3+12*c*a^2*b^4+24*a^2*b^2*c^3+12*a^2*c^5+8*c*b^6+24*b^4*c^3+24*b^2*c^5+8*c^7,
                       -12*(c^2+b^2)*(b^2+c^2+a^2)*a^3)


Similarly, by doing the above procedure, we can obtain a parametric solution of (1) if n is odd and n >= 5.

Thus equation (1) has infinitely many integer solutions if n is odd and n >= 5.

Q.E.D.



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