[#P3124] All nonnegative limits of normalized consecutive-prime gaps

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 \(\lim_{i\to\infty}(p_{n_i+1}-p_{n_i})/\log n_i=C\).

Neutral schematic of a prime number line with consecutive gaps rescaled by a logarithmic ruler. — The objects and operations appearing in All nonnegative limits of normalized consecutive-prime gaps.

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1Context

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.

2Problem setup

Definition 1 (Normalized prime gap). The normalized gap at index \(n\) is \((p_{n+1}-p_n)/\log n\).

Definition 2 (Limit point). A real number \(C\) is a limit point if some subsequence of normalized gaps converges to \(C\).

Remark 1. The question asks whether the finite limit points fill the whole nonnegative real line.

3What 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.

1Status

Saved packet · August 1, 2026

What counts as a solution

Saved packet status (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.^[1][2]

1Packet records

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Record	Kind	Assessment
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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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][2] Relevance to this problem This is the dated publication status for the canonical target All nonnegative limits of normalized consecutive-prime gaps. 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 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. Open full record →
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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 2 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][2] Relevance to this problem Locates the present research frontier immediately below All nonnegative limits of normalized consecutive-prime gaps. 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 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. This leaves the following boundary unresolved: It is unknown whether every finite \(C\ge0\) is a limit point. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target. Related: Claim 1 Open full record →
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.	Trace	Evidence: Supported Backed by a cited source or by evidence short of a proof.
attempt · Route 1 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][2] Relevance to this problem Documents why All nonnegative limits of normalized consecutive-prime gaps was treated as a distinct open target on 2026-08-01. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope No narrower scope recorded Sources Reference 1, Reference 2 What happened The audit ran on 2026-08-01 across source collection and its cited primary literature, Crossref/arXiv or MathSciNet stable records, TheoremDB published, prospecting, formal-conjecture, packet, and retired corpus. It checked the exact statement, its named or normalized variants, recent status evidence, and the controlled TheoremDB corpus. Queries included: - "Erdős Problem #5" normalized prime gaps - "(p_{n+1}-p_n)/log n" limit points - normalized prime gap limit set proof 2025 2026 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. Related: Claim 1 Open full record →
It is unknown whether every finite \(C\ge0\) is a limit point.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 2 It is unknown whether every finite \(C\ge0\) is a limit point. Relevance to this problem Turns the remaining uncertainty in All nonnegative limits of normalized consecutive-prime gaps into a checkable research target. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded What happened Research should address this boundary directly: It is unknown whether every finite \(C\ge0\) is a limit point. A claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations: - 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. Related: Claim 1 Open full record →
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Notes and companion material

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.

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.

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ProblemAll nonnegative limits of normalized consecutive-prime gaps

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2See also

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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Plain text

“All nonnegative limits of normalized consecutive-prime gaps.” TheoremDB. P3124. Problem statement; statement text SHA-256 984c7085100745009b0aa6c37cc9f40ccddc383744e7bbea6543112ca7283edb. https://theoremdb.org/statement/?ref=P3124

BibTeX

@misc{theoremdb-problem-984c7085100745009b0aa6c37cc9f40ccddc383744e7bbea6543112ca7283edb,
  title = {{All nonnegative limits of normalized consecutive-prime gaps}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 984c7085100745009b0aa6c37cc9f40ccddc383744e7bbea6543112ca7283edb},
  url = {https://theoremdb.org/statement/?ref=P3124}
}

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This problem includes 4 records joined by 3 typed links, sourced from erdosproblems.com^[1], current as of August 1, 2026.

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1References

Packet source. 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. ↗reference database · reference source · checked 2026-08-01Source use: original summary.Supplies the maintained formulation, current open-status assessment, and recorded partial results.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.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.
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. ↗ open copy ↗book · primary source · checked 2026-08-01Source 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.

Original TheoremDB statement and summary based on citation-only scholarly sources; no source prose, proof, table, code, or figure is reproduced.

[#P3124] All nonnegative limits of normalized consecutive-prime gaps

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