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

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.

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

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

Current 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]

1Records

Record	Kind	Assessment
	Result	Supported
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 →
	Result	Supported
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 →
	Trace	Supported
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 →
	Trace	Reported
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 →
No records match these filters.

Notes and companion materialContext, examples, and computations

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.

How the 4 records connectTyped relations and evidence flow

ProblemAll nonnegative limits of normalized consecutive-prime gaps

Claim 1Current status and exact unresolved remainderin this packetSupported
Route 1Dated source and duplicate auditchecksSupported
Claim 2Strongest checked neighboring resultinformsSupported
Route 2Work at the unresolved boundaryaddressesReported

2See also

Decidability of positivity for linear recurrence sequencesnumber theory
Erdős Problem 152: isolated sums in Sidon setsnumber theory
Unbounded numbers of integral points on global minimal elliptic curvesnumber theory

How to cite

TheoremDB contributors, “All nonnegative limits of normalized consecutive-prime gaps,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/normalized-prime-gap-limit-set

This page as plain text: normalized-prime-gap-limit-set.md

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

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

1Context

2Problem setup

3What counts as a solution

1Status

1Records

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

Flag this problem

Report a problem

1Context

2Problem setup

3What counts as a solution

1Status

1Records

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance