---

Claim. For every $\epsilon>0$, every sufficiently large $n$ is a sum $n=a+b$ in which neither $a$ nor $b$ has a prime factor exceeding $n^{4/(9\sqrt e)+\epsilon}$; in the notation of Problem 334, $f(n)\ll_\epsilon n^{4/(9\sqrt e)+\epsilon}$, where $4/(9\sqrt e)=0.2695\ldots$. The result is A. Balog, On additive representation of integers, Acta Math. Hungar. 54 (1989), no. 3--4, 297--301, cited as [Ba89] on the problem page, whose commentary records it as the best bound known. The statement is as the zbMATH review of the paper gives it: every $N>N_0(\epsilon)$ is $a+b$ with $a,b>1$ and no prime factor of $ab$ exceeding $N^{4/(9\sqrt e)+\epsilon}$; its proof is not checked here.

Covers. The bound $f(n)\ll_\epsilon n^{4/(9\sqrt e)+\epsilon}$ for every $\epsilon>0$. In particular $f(n)\le n^{1/3}$ for all large $n$, which answers yes the question the site records as Erdős's original one. Not covered: the best function $f$, which the problem asks for and no source determines; the site expects $f(n)\le n^{o(1)}$.

Depends on. No page of this wiki; the claim rests on the cited paper.

Acceptance. Refereed: Acta Mathematica Hungarica 54 (1989), no. 3--4, 297--301, doi:10.1007/BF01952060, the DOI linked above. The site's curator, T. F. Bloom, credits the bound to Balog in the problem page's commentary, but the site labels the problem OPEN (page last edited 2026-04-03), so the credit is not acceptance of the problem and no reviewed evidence is listed.

Dating. The page is dated by the issue month in the publisher's record, September 1989; the day is a placeholder.