[#P32] Cramér's prime-gap conjecture

Problem. If \(p_n\) denotes the \(n\)-th prime, then \(p_{n+1}-p_n=O((\log p_n)^2)\) as \(n\to\infty\).

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Definitions and notation

1Context

Cramér's random model for primes predicts much smaller maximal gaps than unconditional theorems currently provide.

2Definitions

Definition 1 (Big-O here). Big-O here means that some constant C bounds every sufficiently large consecutive-prime gap by C times (log p_n)^2.

Definition 2 (A prime gap). A prime gap is the difference between consecutive prime numbers.

3What counts as a solution

Prove a uniform C(log p)^2 upper bound for all sufficiently large consecutive-prime gaps, or prove that the ratio of a sequence of such gaps to (log p)^2 is unbounded.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Baker, Harman, and Pintz proved unconditionally that every sufficiently large interval [x-x^0.525,x] contains a prime. Consequently p_(n+1)-p_n = O(p_n^0.525). Cramer's O((log p_n)^2) bound remains open. Exact unresolved remainder: Prove a uniform C(log p_n)^2 upper bound for every sufficiently large consecutive-prime gap, or prove that limsup (p_(n+1)-p_n)/(log p_n)^2 is infinite.^[1][2]

1Packet records

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Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: Baker, Harman, and Pintz proved unconditionally that every sufficiently large interval [x-x^0.525,x] contains a prime. Consequently p_(n+1)-p_n = O(p_n^0.525). Cramer's O((log p_n)^2) bound remains open. Exact unresolved remainder: Prove a uniform C(log p_n)^2 upper bound for every sufficiently large consecutive-prime gap, or prove that limsup (p_(n+1)-p_n)/(log p_n)^2 is infinite.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: Baker, Harman, and Pintz proved unconditionally that every sufficiently large interval [x-x^0.525,x] contains a prime. Consequently p_(n+1)-p_n = O(p_n^0.525). Cramer's O((log p_n)^2) bound remains open. Exact unresolved remainder: Prove a uniform C(log p_n)^2 upper bound for every sufficiently large consecutive-prime gap, or prove that limsup (p_(n+1)-p_n)/(log p_n)^2 is infinite.^[1][2] Relevance to this problem For Cramér's prime-gap conjecture, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: Baker, Harman, and Pintz proved unconditionally that every sufficiently large interval [x-x^0.525,x] contains a prime. Consequently p_(n+1)-p_n = O(p_n^0.525). Cramer's O((log p_n)^2) bound remains open. Exact unresolved remainder: Prove a uniform C(log p_n)^2 upper bound for every sufficiently large consecutive-prime gap, or prove that limsup (p_(n+1)-p_n)/(log p_n)^2 is infinite. Open full record →
Prove a uniform C(log p)^2 upper bound for all sufficiently large consecutive-prime gaps, or prove that the ratio of a sequence of such gaps to (log p)^2 is unbounded.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove a uniform C(log p)^2 upper bound for all sufficiently large consecutive-prime gaps, or prove that the ratio of a sequence of such gaps to (log p)^2 is unbounded.^[1] Relevance to this problem For Cramér's prime-gap conjecture, record cramers-prime-gap-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove a uniform C(log p)^2 upper bound for all sufficiently large consecutive-prime gaps, or prove that the ratio of a sequence of such gaps to (log p)^2 is unbounded. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove a uniform C(log p)^2 upper bound for all sufficiently large consecutive-prime gaps, or prove that the ratio of a sequence of such gaps to (log p)^2 is unbounded. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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Notes and companion material

Original intake status. The cited specialist resource presents Cramér's quadratic-logarithmic gap estimate as conjectural. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

Stronger predictions concern the precise limsup constant. This record asks only for the quadratic-logarithmic upper order.

Computational notes

Tables of maximal prime gaps test finite ranges and cannot establish an asymptotic upper bound.

2See also

Improve the upper exponent for the longest Pierce remainder chainanalytic number theory
Riemann hypothesisanalytic number theory
Optimal Gaussian Rank-One Loewner Threshold for the Riemann Theta Kernelanalytic number theory

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Plain text

“Cramér's prime-gap conjecture.” TheoremDB. P32. Problem statement; statement text SHA-256 99e713f86648821a634039a03ddcfcb8336c675fd9994ca6a5dd6da00194ef9f. https://theoremdb.org/statement/?ref=P32

BibTeX

@misc{theoremdb-problem-99e713f86648821a634039a03ddcfcb8336c675fd9994ca6a5dd6da00194ef9f,
  title = {{Cramér's prime-gap conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 99e713f86648821a634039a03ddcfcb8336c675fd9994ca6a5dd6da00194ef9f},
  url = {https://theoremdb.org/statement/?ref=P32}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from primegap-list-project.github.io^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Prime Gap List Project, Frequently Asked Questions, discussion of Cramér's conjecture and computational records, checked 2026-08-01. Prime Gap List Project, discussion of Cramér's conjecture and computational records. ↗website · primary source · checked 2026-07-31Source use: original summary.The cited specialist resource presents Cramér's quadratic-logarithmic gap estimate as conjectural. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at Cramér conjecture and record-gap sections.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Provides maintained computational records and distinguishes them from a proof of the conjectural asymptotic bound.Source named by the research packet.
R. C. Baker, G. Harman, and J. Pintz, “The Difference Between Consecutive Primes, II”. Proceedings of the London Mathematical Society 83(3) (2001), 532-562. DOI 10.1112/plms/83.3.532. Theorem 1, a prime in every sufficiently large interval [x-x^0.525,x]. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Provides the checked unconditional uniform upper bound on consecutive-prime gaps.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P32] Cramér's prime-gap conjecture

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