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There exist infinitely many positive integers - ILL1990 ICE1

Source:

September 18, 2010

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Problem Statement

Prove that there exist infinitely many positive integers

n

such that the number

\frac{1^2+2^2+\cdots+n^2}{n}

is a perfect square. Obviously,

1

is the least integer having this property. Find the next two least integers having this property.