A series of 10584 consecutive integers has a sum that is a perfect cube. What is the smallest possible average of this series?
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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.
Mohammad Riazi-Kermani
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OP has stated that the integers are consecutive not same. – The Demonix _ Hermit Oct 20 '19 at 16:27
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Thanks for the comment, please check the edit – Mohammad Riazi-Kermani Oct 20 '19 at 16:35
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Yup , nearly the same result as I got . Only much neater :) – The Demonix _ Hermit Oct 20 '19 at 16:36