Series of consecutive integers

Question

A series of 10584 consecutive integers has a sum that is a perfect cube. What is the smallest possible average of this series?

Answer 1

We are given $$a+(a+1)+...+(a+10583)=x^3$$ $$ (5292)(2a+10583)=x^3$$ Since $5292=3^3(14)^2$ we have $$2a+10583=14y^3$$ The above equation has no solution because of parity mismatch of the two sides.