Question

Largest perfect square that can divide 2014^3 - 2013^3 + 2012^3 - 2011^3 +...+2^3 - 1^3

Answer 1

–(1³ + 2³ + 3³ +.... + 2014)³ + 2(2³ + 4³ + ... + 2014³)= - (2014 x 2015/ 2)² + 2.2³(1007 x 1008/ 2)²
= –(1007)² (2015)² + 4 . (1007)²(1008)²= –(1007)² [(2016)² – (2015)²]= (1007)² . 4031