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**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.

The paper also includes a less precise result for \(k\)-th powers.

Reviewer: G.Greaves (Cardiff)

### 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
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\textit{P. T. Bateman} et al., Acta Arith. 67, No. 4, 349--380 (1994; Zbl 0815.11048)

### 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.