Sums of distinct squares. (English) Zbl 0815.11048

Let \(N(s)\) denote the largest integer which is not expressible as a sum of \(s\) distinct non-zero squares. Then \(N(s)\) exceeds \(P(s)\), the sum of the first \(s\) non-zero squares. The authors’ results show, in a fairly precise sense, that \(N(s)\) is always close to \(P(s)\). They give an asymptotic formula for the difference \(R(s)=N(s)-P(s)\) having main term \((2s)^{3/2}\), a second term, and an error \(O(s^{9/8})\). They also supply an upper bound for \(R(s)\), essentially by a “greedy” algorithm, which is shown to be best possible (to within a relative error factor of \(\log \log s)\). They make use of results of F. Halter-Koch [Acta Arith. 42, 11-20 (1982; Zbl 0494.10036)] who considered sums of coprime squares.
The paper also includes a less precise result for \(k\)-th powers.


11P05 Waring’s problem and variants


Zbl 0494.10036
Full Text: DOI EuDML