[#P25] Legendre's conjecture

Problem. For every \(n\in\mathbb{Z}_{>0}\), the intersection \(\mathbb{P}\cap(n^2,(n+1)^2)\) is nonempty.

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1Context

The conjecture is one of Landau's classical problems about primes and would give a strong uniform restriction on prime gaps.

2Remarks

Remark 1. Consecutive squares differ by 2n + 1.

Remark 2. The endpoints are composite for n greater than 1, so strict and non-strict versions agree there.

3What counts as a solution

Prove the existence of a prime in every interval (n^2, (n+1)^2), or give a positive integer n and prove that its interval contains no prime.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Unconditionally, every consecutive-square interval contains a 4-almost-prime. Assuming RH, primes occur between consecutive powers x^(2+delta) and (x+1)^(2+delta) in the ranges stated by Chamberland and Straub. Exact unresolved remainder: Prove that every interval (n^2,(n+1)^2) contains a prime.^[1][2]

1Packet records

Show recorded summaries

Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: Unconditionally, every consecutive-square interval contains a 4-almost-prime. Assuming RH, primes occur between consecutive powers x^(2+delta) and (x+1)^(2+delta) in the ranges stated by Chamberland and Straub. Exact unresolved remainder: Prove that every interval (n^2,(n+1)^2) contains a prime.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: Unconditionally, every consecutive-square interval contains a 4-almost-prime. Assuming RH, primes occur between consecutive powers x^(2+delta) and (x+1)^(2+delta) in the ranges stated by Chamberland and Straub. Exact unresolved remainder: Prove that every interval (n^2,(n+1)^2) contains a prime.^[1][2] Relevance to this problem For Legendre's conjecture, this successor gives readable dated status prose and the exact remaining research boundary. 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 packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: Unconditionally, every consecutive-square interval contains a 4-almost-prime. Assuming RH, primes occur between consecutive powers x^(2+delta) and (x+1)^(2+delta) in the ranges stated by Chamberland and Straub. Exact unresolved remainder: Prove that every interval (n^2,(n+1)^2) contains a prime. Open full record →
Prove the existence of a prime in every interval (n^2, (n+1)^2), or give a positive integer n and prove that its interval contains no prime.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove the existence of a prime in every interval (n^2, (n+1)^2), or give a positive integer n and prove that its interval contains no prime.^[1] Relevance to this problem For Legendre's conjecture, record legendres-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove the existence of a prime in every interval (n^2, (n+1)^2), or give a positive integer n and prove that its interval contains no prime. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove the existence of a prime in every interval (n^2, (n+1)^2), or give a positive integer n and prove that its interval contains no prime. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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Notes and companion material

Original intake status. The cited 2026 article calls Legendre's conjecture a famous unsolved problem. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

Recent work proves that every such interval contains an integer with few prime factors. That does not force the integer itself to be prime.

Recorded example 1. Between 10^2 and 11^2 lie the primes 101, 103, 107, 109, and 113.

Computational notes

Verification through a finite n establishes only those intervals.

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

“Legendre's conjecture.” TheoremDB. P25. Problem statement; statement text SHA-256 2ae006d09979b6ae8d905e3ed1fec65bb4dcb113fa52a2a230b8dcd3b5a50ff9. https://theoremdb.org/statement/?ref=P25

BibTeX

@misc{theoremdb-problem-2ae006d09979b6ae8d905e3ed1fec65bb4dcb113fa52a2a230b8dcd3b5a50ff9,
  title = {{Legendre's conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 2ae006d09979b6ae8d905e3ed1fec65bb4dcb113fa52a2a230b8dcd3b5a50ff9},
  url = {https://theoremdb.org/statement/?ref=P25}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from sciencedirect.com^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Adrian W. Dudek and Daniel R. Johnston, Almost primes between all squares, Journal of Number Theory 278 (2026), 726-745. DOI 10.1016/j.jnt.2025.05.009. Adrian W. Dudek and Daniel R. Johnston, Journal of Number Theory 278 (2026), introduction. ↗journal article · primary source · checked 2026-07-31Source use: original summary.The cited 2026 article calls Legendre's conjecture a famous unsolved problem. 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 Abstract and introduction.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.Current unconditional four-almost-prime theorem and open-status statement.Source named by the research packet.
Marc Chamberland and Armin Straub, “Weakening the Legendre Conjecture”. arXiv:2602.22502 (2026). Abstract and main theorem. ↗preprint · primary source · arXiv:2602.22502v1 · checked 2026-08-01Source use: original summary.RH-conditional prime result for consecutive powers above exponent two.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P25] Legendre's conjecture

1Context

2Remarks

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1Lean verification

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1References

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1Context

2Remarks

3What counts as a solution

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1Packet records

Recent contributions

Computational notes

2See also

1References

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Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance