Erdős Problem #208

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.

A positive integer is squarefree when no square of a prime divides it. The ordered sequence of all such integers includes one, and its consecutive gaps refer to that full sequence. [S1] [S2]

Statement recorded in the supplied source.

Let $s_1<s_2<\cdots$ be the sequence of squarefree numbers. Is it true that, for any $\epsilon>0$ and large $n$, $ s_{n+1}-s_n \ll_\epsilon s_n^{\epsilon}? $ Is it true that $ s_{n+1}-s_n \leq (1+o(1))\frac{\pi^2}{6}\frac{\log s_n}{\log\log s_n}? $ [S1]

Transcription note. Only unambiguous escape/formatting damage and nonmathematical serialization debris were repaired in the displayed source statement.

Short English statement

How large can gaps between successive squarefree integers be, both on subpower scales and at the proposed logarithmic scale?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 1993 (EP-208), 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 208, https://www.erdosproblems.com/208. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Erdös, P., Some problems and results in elementary number theory. Publ. Math. Debrecen (1951), 103-109. Bibliographic information from the supplied record.

• S4 Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70. Bibliographic information from the supplied record.

• S5 Filaseta, M. and Trifonov, O., On gaps between squarefree numbers II. J. London Math. Soc. (1992), 215-221. Bibliographic information from the supplied record.

• S6 Granville, Andrew, {$ABC$} allows us to count squarefrees. Internat. Math. Res. Notices (1998), 991--1009. Bibliographic information from the supplied record.