[#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 \(n\to\infty\).
1Context
Cramér's random model for primes predicts much smaller maximal gaps than unconditional theorems currently provide.
2Problem setup
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.
Remark 1. Cramér's random model for primes predicts much smaller maximal gaps than unconditional theorems currently provide.
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
Current 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]
1Records
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
- Riemann hypothesisanalytic number theory
- Improve the upper exponent for the longest Pierce remainder chainanalytic number theory
- Twin prime conjectureprime numbers
How to cite
TheoremDB contributors, “Cramér's prime-gap conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/cramers-prime-gap-conjectureThis page as plain text: cramers-prime-gap-conjecture.md
This problem includes 2 records joined by 2 typed links, sourced from primegap-list-project.github.io[1], current as of July 31, 2026.
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.