Residue Class Multiple Coverage

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 each prime $p\leq N$ choose one residue $a_p\in{\mathbb{Z}}/p{\mathbb{Z}}$. The coverage multiplicity of $m\in[N]$ is $c(m)=\#\{p\leq N:p\text{ prime},\ m\equiv a_p\pmod p\}$. The question is whether, for all sufficiently large $N$, a choice can achieve $\min_{m\in[N]}c(m)\geq10$. Different primes, not repeated use of a congruence, contribute to this count. [S1] [S2]

Statement recorded in the supplied source.

Can we pick residue classes $a_p \pmod p$, one for each prime $p \leq N$, such that every integer $\leq N$ lies in at least 10 of them? [S1]

Short English statement

Can one residue class for each available prime be chosen so that every integer in the initial interval is covered at least ten times?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 175 (GREEN-087), 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 #689 and discussion. https://www.erdosproblems.com/689. Bibliographic information from the supplied record.

• S3 Erdős Problems #689 discussion thread, proposed-proof posts dated April--June 2026. https://www.erdosproblems.com/forum/thread/689. Bibliographic information from the supplied record.

• S4 B. Green, 100 Open Problems, Problem 45, 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.