How large can gaps between consecutive squarefree numbers be? (Erdős #208)
Statement
Let $s_1<s_2<\cdots$ be the squarefree numbers. Two questions about the maximal gap $s_{n+1}-s_n$: (1) Is it true that for every $\epsilon>0$ one has $s_{n+1}-s_n \ll_\epsilon s_n^{\epsilon}$ for all large $n$? (2) Is it true that $$s_{n+1}-s_n \le (1+o(1))\,\frac{\pi^2}{6}\,\frac{\log s_n}{\log\log s_n}?$$
Acceptance. FULLY RESOLVES: prove question (1) unconditionally — $s_{n+1}-s_n\ll_\epsilon s_n^{\epsilon}$ for every $\epsilon>0$ — and/or prove the sharp bound (2); a complete proof is required (Lean/Coq preferred, else a full written proof). Because (1) is already known under abc [Gr98], only an unconditional proof or a genuinely new conditional route counts as fully resolving. ADVANCES (each checkable): improve the unconditional exponent in $s_{n+1}-s_n\ll s_n^{\theta}$ strictly below the best value stated in the background ($\theta=1/5-c$), with proof; or produce a certified table of maximal squarefree gaps up to a stated height with a new record gap, delivering the sieve code and an exhaustiveness certificate that tests the conjectured constant $\pi^2/6$. Deliver the proof with the exponent obtained, or the gap-record computation plus certificate.
Background
Asked by Erdős [Er51; Er61; Er65b; Er79; Er81h, p.176]. Listed as open on erdosproblems.com/208 (fetched 2026-07-13, status 'open', tagged 'number theory'); a Lean 4 formalisation exists in google-deepmind/formal-conjectures. The bound in (2) would be best possible: Erdős [Er51] showed there are infinitely many $n$ with $s_{n+1}-s_n > (1+o(1))\frac{\pi^2}{6}\frac{\log s_n}{\log\log s_n}$, matching the conjectured upper bound. Erdős [Er79] speculated one might even have $s_{n+1}-s_n \ll \log s_n$ but was 'very doubtful'. Unconditional upper bounds on the gap: Filaseta and Trifonov [FiTr92] proved $s_{n+1}-s_n \ll s_n^{1/5+o(1)}$, and Pandey [Pa24] improved the exponent to $1/5-c$ for an explicit constant $c>0$ — still far from the conjectured $(\log s_n)^{1+o(1)}$ size. Granville [Gr98] showed the near-optimal bound $s_{n+1}-s_n \ll_\epsilon s_n^{\epsilon}$ of question (1) follows from the abc conjecture. See the companion moment problem Erdős #145 (erdosproblems.com/145) and the more general Erdős #1101 (erdosproblems.com/1101). No Erdős prize is recorded. Attacker's tool: the Filaseta–Trifonov gap method (exponential sums / counting squarefull numbers in short intervals) to push the exponent below $1/5-c$, together with a segmented squarefree sieve to locate record gaps and test the conjectured extremal constant $\pi^2/6$ numerically.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #208 (T. F. Bloom) | website |
| REF-02 | OEIS A005117 — the squarefree numbers | website |
| REF-03 | OEIS A076259 — differences between consecutive squarefree numbers | website |
| REF-04 | Lean 4 formalisation — google-deepmind/formal-conjectures #208 | website |
Investigations · 0
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