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N^2 as sum of the squares of n consecutive positive intege

Source: Singapore Open Math Olympiad 2001 2nd Round p4 SMO

April 2, 2020
consecutiveSum of SquaresSumnumber theory

Problem Statement

A positive integer nn is said to possess Property (AA) if there exists a positive integer NN such that N2N^2 can be written as the sum of the squares of nn consecutive positive integers. Is it true that there are infinitely many positive integers which possess Property (AA)? Justify your answer. (As an example, the number n=2n = 2 possesses Property (AA) since 52=32+425^2 = 3^2 + 4^2).
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