Maximal Covering Interval

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 real $x\geq2$, let $Y(x)$ be the maximum integer $y$ for which residues $a_p\pmod p$, one for each prime $p\leq x$, cover every integer in $[y]$. This is an extremal choice of residues, not the particular choice $a_p=0$. The question requests sharp bounds or an asymptotic description of $Y(x)$. [S1] [S2]

Statement recorded in the supplied source.

What is the largest $y$ for which one may cover the interval $[y]$ by residue classes $a_p \pmod p$, one for each prime $p \leq x$? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Improve either side substantially, in particular prove the conjectural x^{1+o(1)} upper bound or a contrary superlinear-power lower bound. [S1]

Short English statement

How long an initial interval can be covered when one freely chooses a single residue class for each prime up to a given bound?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 176 (GREEN-088), 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. Ford, B. Green, S. Konyagin, J. Maynard, T. Tao, Long gaps between primes, J. Amer. Math. Soc. 31 (2018), 65-105. https://arxiv.org/abs/1412.5029. Bibliographic information from the supplied record.

• S3 H. Iwaniec, On the problem of Jacobsthal, Demonstratio Math. 11 (1978), 225-232,

  \nolinkurl{doi:10.1515/dema-1978-0121.}

  https://doi.org/10.1515/dema-1978-0121. Bibliographic information from the supplied record.

• S4 B. Green, 100 Open Problems, Problem 46, 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.

• S5 T. F. Bloom, Erdős Problem #970. https://www.erdosproblems.com/970. Bibliographic information from the supplied record.