Erdős Problem #786

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.

For an integer set $A$, its natural density is $\lim_{X\to\infty}|A\cap[1,X]|/X$, when this limit exists. Upper and lower densities use limit superior and limit inferior. A request for density is not automatically satisfied by positive upper density alone. [S1] [S2]

Statement recorded in the supplied source.

Let $\epsilon>0$. Is there some set $A\subset \mathbb{N}$ of density $>1-\epsilon$ such that $a_1\cdots a_r=b_1\cdots b_s$ with $a_i,b_j\in A$ can only hold when $r=s$? Similarly, can one always find a set $A\subset\{1,\ldots,N\}$ with this property of size $\geq (1-o(1))N$? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Verify the intended factor-repetition convention from the original source, then attach the corresponding result. [S1]

Short English statement

Can sets of density arbitrarily close to one force equal products of their elements to have the same number of factors?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2303 (EP-786), 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 786, https://www.erdosproblems.com/786. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189. Bibliographic information from the supplied record.