Bateman, Paul T.; Hildebrand, Adolf J.; Purdy, George B. Sums of distinct squares. (English) Zbl 0815.11048 Acta Arith. 67, No. 4, 349-380 (1994). 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. Reviewer: G.Greaves (Cardiff) Cited in 1 Document MSC: 11P05 Waring’s problem and variants Keywords:greedy algorithm; largest integer not expressible as sum of distinct squares; sum of first nonzero squares; asymptotic formula; difference; upper bound Citations:Zbl 0494.10036 × Cite Format Result Cite Review PDF Full Text: DOI EuDML Online Encyclopedia of Integer Sequences: Numbers that are the sum of 5 distinct nonzero squares. Numbers that are not the sum of 5 distinct squares. Numbers that are not the sum of 4 distinct nonzero squares. Numbers that are not the sum of 5 distinct nonzero squares. Largest number not the sum of n distinct nonzero squares. Numbers that are the sum of exactly 6 distinct nonzero squares. Numbers that are the sum of exactly 7 distinct nonzero squares. Numbers that are the sum of exactly 8 distinct nonzero squares. Length of longest sequence of distinct nonzero squares summing to n, or 0 if no such sequence exists.