Erdős Problem #334

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

A positive integer is $y$-smooth if all its prime factors are at most $y$, with 1 smooth for every $y\geq1$. For targets $n\geq2$, the pointwise optimal bound is $\min_{1\leq a<n}\max(P(a),P(n-a))$, taking $P(1)=1$. [S1] [S2]

Statement recorded in the supplied source.

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 $p>f(n)$.) [S1]

Scope and interpretation.

With positive summands the target n=1 is impossible. The displayed optimization is therefore read for n at least two; the source is an asymptotic smooth-summand question. [S1]

Short English statement

How small can the largest allowed prime factor be while still permitting each positive target to be written as a sum of two smooth positive integers?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2057 (EP-334), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 T. F. Bloom, Erdős Problems, Problem 334, https://www.erdosproblems.com/334. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Balog, A., On additive representation of integers. Acta Math. Hungar. (1989), 297-301. Bibliographic information from the supplied record.