Products of Primes Modulo p

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 a prime $p$, write $\mathcal P_p=\{q<p:q\text{ is prime}\}$. The question is whether $\mathcal P_p\mathcal P_p=({\mathbb{Z}}/p{\mathbb{Z}})^\times$ for every sufficiently large $p$, where products are reduced modulo $p$ and the same prime may be used twice. The threshold must be independent of the requested nonzero residue. [S1] [S2]

Statement recorded in the supplied source.

Let $p$ be a large prime, and let $A$ be the set of all primes less than $p$. Is every $x \in \{1, \dots, p-1\}$ congruent to some product $a_1a_2$ modulo $p$? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Show that all p-1 nonzero classes modulo every sufficiently large prime p are represented by two primes below p, or find an obstruction. [S1]

Short English statement

For every sufficiently large prime modulus, can every nonzero residue be represented by the product of two smaller primes?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 123 (GREEN-035), 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 K. Matomäki, J. Teräväinen, Products of primes in arithmetic progressions, J. Reine Angew. Math. 808 (2024), 193-240,

  \nolinkurl{doi:10.1515/crelle-2023-0096.}

  https://arxiv.org/abs/2301.07679. Bibliographic information from the supplied record.

• S3 B. Green, 100 Open Problems, Problem 62, 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.