Multiplicatively Closed Set Density

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.

Define $A_0=\{2,3\}$ and $A_{j+1}=A_j\cup\{ab-1:a,b\in A_j\}$. Then $A=\bigcup_{j\geq0}A_j$ is the smallest set in the statement. The factors are not required to be distinct. Natural density, when it exists, is $\lim_{N\to\infty}|A\cap[N]|/N$. The phrase positive density" does not by itself assert a particular numerical value. [S1] [S2]

Statement recorded in the supplied source.

Let $A$ be the smallest set containing 2 and 3, and closed under the operation $a_1a_2 - 1$ (if $a_1, a_2 \in A$, then $a_1a_2 - 1 \in A$). Does $A$ have positive density? [S1]

Scope and interpretation.

The supplied statement permits equal factors. A version requiring two distinct factors is a different closure process and is not silently substituted. [S1]

Short English statement

Does repeatedly adjoining one less than a product, starting from two and three, eventually produce a positive proportion of the integers?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 124 (GREEN-036), 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 Problem #424. https://www.erdosproblems.com/424. Bibliographic information from the supplied record.

• S3 B. Green, 100 Open Problems, Problem 63, version. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf. The author-hosted document was inspected on 27 September 2026; the archive’s local code is not a problem-number locator in this paper.