SMO 2001 Problem 4

A positive integer $n$ is said to possess Property ( $A$ ) if there exists a positive integer $N$ such that $N^2$ can be written as the sum of the squares of $n$ consecutive positive integers. Is it true that there are infinitely many positive integers which possess Property ( $A$ )? Justify your answer. (As an example, the number $n = 2$ possesses Property ( $A$ ) since $5^2 = 3^2 + 4^2$ ).