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Statement

Notation (printed p. 4): $n_2(k)$ is the largest integer such that all of $0,1,2,\ldots,n_2(k)$ are sums of two elements of some system of $k$ non-negative integers (the zero counted); the least $k$ with $n_2(k)\ge n$ is the size $g(n)$ of a minimal 2-basis for $n$.

Conjecture (printed p. 9, unlabeled, quoted with its lead-in): "Die Verbesserung, die die Basis (11) im Vergleich zur Basis (6) bringt, erstreckt sich nur auf das in $k$ lineare Glied von (9). Es ist zu vermuten, daß

$$n_2(k)=\frac{k^2}4+O(k)$$

ist."

It closes § 1 after the two constructions: (9) $n_2(k)\ge\frac{k^2}4+ \frac32k-\gamma$ from the basis (6) of Satz 2, and (15)--(16) from the basis (11) of Satz 4, which improve only the linear term (p. 8). The counting bound (2) gives $n_2(k)\le\frac{k^2+k}2-1$, so the conjecture asserts that the constant $\frac14$ of the constructions, not the $\frac12$ of the count, is the truth.

In the problem's notation. Since $g(n)=\min\{k:n_2(k)\ge n\}$, the conjecture is $g(n)=2\sqrt n+O(1)$. Erdős 1973 reports it as "Rohrbach conjectured $g(n)=2\sqrt n+o(1)$" (printed p. 131 of Erdős 1973, as printed there), and the site's Problem 791 asks its asymptotic form, "is it true that $g(n)\sim2n^{1/2}$?". Each of the three forms implies $\lim n_2(k)/k^2=\frac14$, and each is refuted by Mrose's equation (3), $n_2(k)\ge\frac87(\frac k2)^2+O(k)$ (his $k$ counting positive elements, which changes no ratio), so that $\liminf n_2(k)/k^2\ge\frac27>\frac14$ and $\limsup g(n)/\sqrt n\le\sqrt{7/2}<2$; Kohonen 2017 raises the $\liminf$ to $\frac{85}{294}$.

Source. H. Rohrbach, Ein Beitrag zur additiven Zahlentheorie, Math. Z. 42 (1937), 1--30, doi:10.1007/BF01160061; the conjecture on printed p. 9 = PDF p. 9, the definition of $n_2(k)$ on printed p. 4 = PDF p. 4 of the publisher's scan, read on the page images. The artifact is identified in the source digest.

Read depth. Claims checked: the sentence and its lead-in, and the definition of $n_2(k)$, were read clause by clause on the page images. A conjecture carries no proof; the constructions that motivated it are recorded on satz_3. Nothing here is independently reviewed.

Proof pointer

None; a conjecture. The paper's evidence for it is that the second construction (11) improved only the linear term of (9) (pp. 8--9).

Dependencies

None.

Bears on

• Problem 791: the origin of the "in particular" question $g(n)\sim2n^{1/2}$, in its original and stronger form $g(n)=2\sqrt n+O(1)$; answered in the negative by Mrose 1979, as the problem page records.