# P25: Legendre's conjecture

- ID: `P25`
- Reference: `legendres-conjecture`
- Page: https://theoremdb.org/statements/P25
- Record maturity: Reviewed problem with recorded work

## Problem

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

### Context

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

### Problem setup

- **Definition (Consecutive squares differ by 2n + 1).** Consecutive squares differ by 2n + 1.
- **Definition (The endpoints are composite for n greater than 1, so strict and non-strict versions agree there).** The endpoints are composite for n greater than 1, so strict and non-strict versions agree there.
- **Remark.** The conjecture is one of Landau's classical problems about primes and would give a strong uniform restriction on prime gaps.

### What 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.

## Status

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](#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: 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.

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.

### Background and intake notes

- Original intake status: The cited 2026 article calls Legendre's conjecture a famous unsolved problem. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The statement and status were checked against the cited article on 2026-07-22.
- 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: Between 10^2 and 11^2 lie the primes 101, 103, 107, 109, and 113.

### Open directions

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

### Computational notes

- Verification through a finite n establishes only those intervals.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `legendres-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>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 https://www.sciencedirect.com/science/article/pii/S0022314X25001702
   - Also cited at Abstract and introduction
   - Also cited at Editorial research route recorded 2026-07-31
   - journal_article; primary source; checked 2026-07-31
   - Source 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.
   - 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.
2. <a id="reference-2"></a>Marc Chamberland and Armin Straub, “Weakening the Legendre Conjecture”. arXiv:2602.22502 (2026). Abstract and main theorem https://arxiv.org/abs/2602.22502
   - preprint; primary source; arXiv:2602.22502v1; checked 2026-08-01
   - Source use: original_summary
   - RH-conditional prime result for consecutive powers above exponent two.