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

We rewrite the equation as a + (a+1) + (a+2) + ... + (a+10583) = x^3. Adding them all up and factoring, we get (5292)(2a+10583) = x^3. We can factor 5292 as 3^3 * 14^2. That means that 2a+10583 = 14y^3, which has no solution because of the parity mismatch.