# P32: Cramér's prime-gap conjecture

- ID: `P32`
- Reference: `cramers-prime-gap-conjecture`
- Page: https://theoremdb.org/statements/P32
- Record maturity: Reviewed problem with recorded work

## Problem

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

### Context

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

### Problem setup

- **Definition (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 (A prime gap).** A prime gap is the difference between consecutive prime numbers.
- **Remark.** Cramér's random model for primes predicts much smaller maximal gaps than unconditional theorems currently provide.

### What 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.

## Status

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](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- Original intake status: The cited specialist resource presents Cramér's quadratic-logarithmic gap estimate as conjectural. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The standard big-O formulation and current status were checked on 2026-07-22.
- Stronger predictions concern the precise limsup constant. This record asks only for the quadratic-logarithmic upper order.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Computational notes

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

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `cramers-prime-gap-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://primegap-list-project.github.io/faq/
   - Also cited at Cramér conjecture and record-gap sections
   - Also cited at Editorial research route recorded 2026-07-31
   - website; primary source; checked 2026-07-31
   - Source 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.
   - 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.
2. <a id="reference-2"></a>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] https://doi.org/10.1112/plms/83.3.532
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Provides the checked unconditional uniform upper bound on consecutive-prime gaps.