# P5: Erdős-Straus conjecture

- ID: `P5`
- Reference: `erdos-straus-conjecture`
- Page: https://theoremdb.org/statements/P5
- Record maturity: Reviewed problem with recorded work

## Problem

For every integer \(n\ge 2\), there exist positive integers \(x,y,z\) such that \(\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\).

### Context

This asks whether every fraction 4/n has a representation as a sum of three positive unit fractions.

### Problem setup

- **Definition (A unit fraction).** A unit fraction is a fraction whose numerator is 1 and whose denominator is a positive integer.
- **Definition (The denominators x, y, and z need not be distinct).** The denominators x, y, and z need not be distinct.
- **Remark.** This asks whether every fraction 4/n has a representation as a sum of three positive unit fractions.

### What counts as a solution

- Give a proof that a positive-integer decomposition exists for every n at least 2, or exhibit an n and prove that no positive integers x, y, and z satisfy the equation.

## Status

Unresolved in this packet after the dated source check. Strongest checked result: Xu reduces attention to primes congruent to 1 modulo 24, parameterizes tame solution classes, and covers almost all tested tame primes in a stated finite range. The universal decomposition remains open. Exact unresolved remainder: Prove a positive three-unit-fraction decomposition for every integer n at least 2, or give an n and prove that no such decomposition exists. [1](#reference-1)

## 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: Xu reduces attention to primes congruent to 1 modulo 24, parameterizes tame solution classes, and covers almost all tested tame primes in a stated finite range. The universal decomposition remains open. Exact unresolved remainder: Prove a positive three-unit-fraction decomposition for every integer n at least 2, or give an n and prove that no such decomposition exists.

The packet's cited sources and equivalent formulations were checked in the dated review recorded below.

Strongest checked result: Xu reduces attention to primes congruent to 1 modulo 24, parameterizes tame solution classes, and covers almost all tested tame primes in a stated finite range. The universal decomposition remains open.

Exact unresolved remainder: Prove a positive three-unit-fraction decomposition for every integer n at least 2, or give an n and prove that no such decomposition exists.

### Background and intake notes

- Original intake status: The cited 2026 research preprint identifies the conjecture as open. 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 2026 preprint on 2026-07-22.
- Many congruence classes and large finite ranges are known. Consult the cited paper and its references before claiming a new case or family.

- Recorded example: For n = 2, one has 4/2 = 1/1 + 1/2 + 1/2.

### Open directions

- **Route 1** (reported): Give a proof that a positive-integer decomposition exists for every n at least 2, or exhibit an n and prove that no positive integers x, y, and z satisfy the equation. [1](#reference-1)

### Computational notes

- A finite verification establishes cases through its bound and leaves the universal statement open.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-straus-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>Xiaoping Xu, “Congruence Classes of Supporting the Erdös-Straus Conjecture I: Tame Solutions”. arXiv:2605.23601 (2026). Xiaoping Xu, arXiv:2605.23601, introduction and abstract https://arxiv.org/abs/2605.23601
   - Also cited at abstract, reduction to primes 1 modulo 24, and tame congruence classes
   - Also cited at Editorial research route recorded 2026-07-31
   - preprint; primary source; arXiv:2605.23601, checked 2026-07-31; checked 2026-07-31
   - Source use: original_summary
   - The cited 2026 research preprint identifies the conjecture as open. 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.
   - Provides current congruence families and bounded evidence while stating the universal problem as open.
   - Source named by the research packet.