Erdős Problem #676

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.

The desired expression has $a\in{\mathbb{N}}$, prime $p$, and integer remainder $0\leq b<p$. Equivalently, the residue of $n$ modulo $p^2$ belongs to $\{0,\ldots,p-1\}$ and $n\geq p^2$. [S1] [S2]

Statement recorded in the supplied source.

Is every sufficiently large integer of the form $ ap^2+b $ for some prime $p$ and integer $a\geq 1$ and $0\leq b<p$? [S1]

Short English statement

Is every sufficiently large integer less than one prime above a positive multiple of the square of that same prime?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2248 (EP-676), 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 676, https://www.erdosproblems.com/676. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Paul Erdős, Some unconventional problems in number theory, Mathematics Magazine 52 (1979), 67-70. https://doi.org/10.1080/0025570X.1979.11976756. Bibliographic information from the supplied record.

• S4 Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. Bibliographic information from the supplied record.