# P3124: All nonnegative limits of normalized consecutive-prime gaps

- ID: `P3124`
- Reference: `normalized-prime-gap-limit-set`
- Page: https://theoremdb.org/statements/P3124
- Record maturity: Reviewed problem with recorded work

## Problem

Let \(p_n\) be the \(n\)-th prime. Prove or disprove that for every real \(C\ge 0\) there is a strictly increasing sequence \((n_i)_{i\ge1}\) such that \(\lim_{i\to\infty}(p_{n_i+1}-p_{n_i})/\log n_i=C\).

### Context

Known frontier: The maintained survey records \(0\) and \(\infty\) as limit points, arbitrarily large finite limit points, an interval \([0,c]\), positive density of the limit set, and bounded gaps in that set.

Open boundary: It is unknown whether every finite \(C\ge0\) is a limit point.

### Problem setup

- **Definition (Normalized prime gap).** The normalized gap at index \(n\) is \((p_{n+1}-p_n)/\log n\).
- **Definition (Limit point).** A real number \(C\) is a limit point if some subsequence of normalized gaps converges to \(C\).
- **Remark.** The question asks whether the finite limit points fill the whole nonnegative real line.

### What counts as a solution

- For a proof, construct or establish a convergent subsequence for every \(C\ge0\).
- For a disproof, exhibit a specific \(C\ge0\) and prove that no normalized-gap subsequence converges to it.

## Status

OPEN as checked on 2026-08-01. Strongest checked neighboring result: The maintained survey records \(0\) and \(\infty\) as limit points, arbitrarily large finite limit points, an interval \([0,c]\), positive density of the limit set, and bounded gaps in that set. Exact unresolved remainder: It is unknown whether every finite \(C\ge0\) is a limit point. [1](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (Current status and exact unresolved remainder).** OPEN as checked on 2026-08-01. Strongest checked neighboring result: The maintained survey records \(0\) and \(\infty\) as limit points, arbitrarily large finite limit points, an interval \([0,c]\), positive density of the limit set, and bounded gaps in that set. Exact unresolved remainder: It is unknown whether every finite \(C\ge0\) is a limit point.

The problem was checked as open on 2026-08-01.

The strongest neighboring result found in the cited sources is: The maintained survey records \(0\) and \(\infty\) as limit points, arbitrarily large finite limit points, an interval \([0,c]\), positive density of the limit set, and bounded gaps in that set.

The exact unresolved remainder is: It is unknown whether every finite \(C\ge0\) is a limit point.

A complete resolution must meet the following acceptance conditions:
- For a proof, construct or establish a convergent subsequence for every \(C\ge0\).
- For a disproof, exhibit a specific \(C\ge0\) and prove that no normalized-gap subsequence converges to it.

### Background and intake notes

- Original intake status: OPEN as checked on 2026-08-01. Strongest checked neighboring result: The maintained survey records \(0\) and \(\infty\) as limit points, arbitrarily large finite limit points, an interval \([0,c]\), positive density of the limit set, and bounded gaps in that set. Exact unresolved remainder: It is unknown whether every finite \(C\ge0\) is a limit point.
- The release review checked 2 structured sources on 2026-08-01.
- Equivalent-formulation queries: "Erdős Problem #5" normalized prime gaps; "(p_{n+1}-p_n)/log n" limit points; normalized prime gap limit set proof 2025 2026
- Strongest checked neighboring result: The maintained survey records \(0\) and \(\infty\) as limit points, arbitrarily large finite limit points, an interval \([0,c]\), positive density of the limit set, and bounded gaps in that set.
- Exact unresolved remainder: It is unknown whether every finite \(C\ge0\) is a limit point.

### Other known results

- **Claim 2** (supported): The maintained survey records \(0\) and \(\infty\) as limit points, arbitrarily large finite limit points, an interval \([0,c]\), positive density of the limit set, and bounded gaps in that set. [1](#reference-1) [2](#reference-2)

### Prior approaches

- **Route 1** (supported): The exact formulation, named variants, 2025–2026 updates, and repository-wide semantic duplicates were checked on 2026-08-01. The source collection still marks the stated remainder open. Living-database status remains subject to later literature not indexed there. [1](#reference-1) [2](#reference-2)

### Open directions

- **Route 2** (reported): It is unknown whether every finite \(C\ge0\) is a limit point.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `normalized-prime-gap-limit-set`, 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>Thomas F. Bloom, Erdős Problem #5, Erdős Problems database (living entry), accessed 2026-08-01. Problem #5, OPEN banner, statement, remarks, and bibliography. Problem #5, OPEN banner, statement, remarks, and bibliography https://www.erdosproblems.com/5
   - Also cited at Thomas F. Bloom, Erdős Problem #5, Erdős Problems database (living entry), accessed 2026-08-01. Problem #5, OPEN banner, statement, remarks, and bibliography
   - reference_database; reference source; checked 2026-08-01
   - Source use: original_summary
   - Supplies the maintained formulation, current open-status assessment, and recorded partial results.
   - Source used to assess the problem's recorded status.
   - For All nonnegative limits of normalized consecutive-prime gaps: This is the dated publication status for the canonical target All nonnegative limits of normalized consecutive-prime gaps.
   - Source named by the research packet.
2. <a id="reference-2"></a>P. Erdős, “Some Problems On The Distribution Of Prime Numbers,” in Teoria dei numeri, C.I.M.E. Summer Schools 5 (1955), 79–88. pp. 79–88, problem on limit points of consecutive-prime gaps https://doi.org/10.1007/978-3-642-10892-1_3
   - book; primary source; checked 2026-08-01
   - Open copy: https://users.renyi.hu/~p_erdos/1955-12.pdf
   - Source use: original_summary
   - Records an original formulation or early published statement of the problem.
   - Source used to assess the problem's recorded status.
   - For All nonnegative limits of normalized consecutive-prime gaps: Records an original formulation or early published statement of the problem.