Problem 334

Find the best function $f(n)$ such that every $n$ can be written as $n=a+b$ where both $a,b$ are $f(n)$ -smooth (that is, are not divisible by any prime $p>f(n)$ .)

Source: erdosproblems.com/334

Number theory

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Open

No claim settles this problem.

Open, the site's label (OPEN; page last edited 2026-04-03). The case the site records as Erdős's original question, whether $f(n)\le n^{1/3}$ , is answered yes by Balog [Ba89], recorded as the accepted partial claim Balog's bound with exponent $4/(9\sqrt e)$ ; the best function $f$ is not determined, so the problem stays open.

Ba89: Balog, A., On additive representation of integers. Acta Math. Hungar. (1989), 297-301.
Er76e: Erdős, P., Problems and results on consecutive integers. Publ. Math. Debrecen (1976), 271-282. Library home: Erdos 1976.
Er82d: Erdős, P., Some new problems and results in number theory. Number theory (Mysore, 1981), Lecture Notes in Math. 938 (1982), 50-74. Library home: Erdős 1982.
ErGr80: Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). Library home: Erdős and Graham 1980.

Problem 334

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