Erdős Problem #279

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.

Choose representatives $0\leq a_p<p$; otherwise changing a representative changes the cutoff $t\geq k$. The question is about the union of these truncated prime progressions, not merely about their untruncated residue classes. [S1] [S2]

Statement recorded in the supplied source.

Let $k\geq 3$. Is there a choice of congruence classes $a_p\pmod{p}$ for every prime $p$ such that all sufficiently large integers can be written as $a_p+tp$ for some prime $p$ and integer $t\geq k$? [S1]

Short English statement

Can one residue modulo each prime cover every sufficiently large integer even after omitting a prescribed number of initial terms of each progression?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2027 (EP-279), 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 279, https://www.erdosproblems.com/279. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Paul Erdős and Ronald L. Graham, Old and New Problems and Results in Combinatorial Number Theory (1980), p. 29. https://combinatorica.hu/~p_erdos/1979-07.pdf. Bibliographic information from the supplied record.