Squares Expressible as Sum of Consecutive Squares
Laurent Beeckmans · American Mathematical Monthly · 1994
Little has been published about this problem but a good account of it can be found in [6], where it is shown that S is infinite and has density zero. Moreover, [6] contains a table giving all the elements of S less than 73. In the present paper, we extend this table to include all the elements of S less than 1000. But our main purpose is first to give necessary conditions that k must satisiFy in order to belong to S. Then we will describe a general method to find the squares which are sums of k consecutive squares, for any given k, this method being an application of the theory of Pell's equation. Furthermore, we will show that if k belongs to S, then there exist infinitely many squares that can be written as the sum of k consecutive squares if and only if k itself is not a square. As usual, we will write Pallk iffpalk but p+1+ k; cr alwayswill denote a strictly positive integer.