---

Claim. H. Rohrbach, Ein Beitrag zur additiven Zahlentheorie, Math. Z. 42 (1937), 1--30. A system of $k$ non-negative integers, the zero counted, is a $2$-basis for $n$ if every integer $0,1,\ldots,n$ is a sum of two of its elements; the least such $k$ is the $g(n)$ of Problem 791 (p. 4). The paper proves:

• Satz 3 (p. 5): a minimal $2$-basis for $n>1$ has fewer than $2\sqrt n$ elements, that is $g(n)<2\sqrt n$, by the explicit basis (6) of Satz 2;
• the Folgerung to Satz 6 (pp. 14--15): every $2$-basis of $k$ elements for $n$ has $n\le k^2/2$, so $g(n)^2\ge2n$ whenever $g(n)\ge5$;
• inequality (47) (p. 18): every $2$-basis of $k$ elements for $n$ has $n<0.4992\,k^2$ once $k$ is large enough.

Since $g(n)\ge\sqrt{2n}\to\infty$, (47) applies to a minimal basis for all large $n$ and gives $n<0.4992\,g(n)^2$, that is $g(n)^2>(2.0032\ldots)n$ for large $n$. Together with Satz 3 this is the site's $(2+c)n\le g(n)^2\le4n$, with $c=0.0032$.

Covers. The bounds $g(n)<2\sqrt n$ for $n>1$ and $g(n)^2>(2.0032\ldots)n$ for large $n$. Not covered: the estimate of $g(n)$ beyond these constants. The paper's conjecture $n_2(k)=k^2/4+O(k)$ (p. 9) is the problem's "in particular" question and is not a claim; it is refuted on Hämmerer and Hofmeister's claim page and Mrose's.

Depends on.

• Rohrbach 1937, Satz 3
• Rohrbach 1937, inequality (47)

Acceptance. Refereed: the paper is published in Mathematische Zeitschrift (Crossref: 1937-12), which dates this page. The site's curator, Thomas F. Bloom, attributes these bounds to Rohrbach in the problem's commentary, but the site labels the problem OPEN, so the attribution is not listed as reviewed. The proof of Satz 3 is followed on the result page; the proofs of the lower bounds in §§ 3--5 are not checked in this corpus.