Integers which are sums of consecutive squares (Q1342740)

It is shown that the set \(T\) consisting of all sums of consecutive squares forms a basis (for \(\{1, 2,\dots\}\)) of the exact order 3. For the counting function \(\tau(s)\) of \(T\) the upper bound \(\tau(s)< 1.84 s^{2/3}\) is obtained estimating the number of all sums \(\sum_{j=0}^{n-1} (x+j)^ 2\leq s\) for positive integers \(x\). Some \(t\in T\) have different representations, e.g. \(25= 5^ 2= 3^ 2+ 4^ 2\), \(365= 13^ 2+ 14^ 2= 10^ 2+ 11^ 2+ 12^ 2\). This leads to the diophantine problem solving \(\sum_{j=0}^{n-1} (x+j)^ 2= \sum_{j=0}^{m-1} (y+j)^ 2\) which generalizes an old and yet open problem of Lucas.