[#P5] Erdős-Straus conjecture

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

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Definitions and notation

1Context

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

2Problem setup

Definition 1 (A unit fraction). A unit fraction is a fraction whose numerator is 1 and whose denominator is a positive integer.

Remark 1. The denominators x, y, and z need not be distinct.

3What 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.

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: 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]

1Packet records

Show recorded summaries

Record	Kind	Assessment
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.	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: 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] Relevance to this problem For Erdős-Straus 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 Details 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. Open full record →
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.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 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] Relevance to this problem For Erdős-Straus conjecture, record erdos-straus-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: 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. 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. 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. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
No records match these filters.

Notes and companion material

Original intake status. The cited 2026 research preprint identifies the conjecture as open. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

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 1. For n = 2, one has 4/2 = 1/1 + 1/2 + 1/2.

Computational notes

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

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

“Erdős-Straus conjecture.” TheoremDB. P5. Problem statement; statement text SHA-256 12120aea8ba7a5b3a15a32b39964d9878e37b81457826a19af00a59cba67391d. https://theoremdb.org/statement/?ref=P5

BibTeX

@misc{theoremdb-problem-12120aea8ba7a5b3a15a32b39964d9878e37b81457826a19af00a59cba67391d,
  title = {{Erdős-Straus conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 12120aea8ba7a5b3a15a32b39964d9878e37b81457826a19af00a59cba67391d},
  url = {https://theoremdb.org/statement/?ref=P5}
}

Plain text: Built Markdown snapshot

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

1Lean verification

1References

Packet source. 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. ↗preprint · primary source · arXiv:2605.23601, checked 2026-07-31 · checked 2026-07-31Source 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.Also cited at abstract, reduction to primes 1 modulo 24, and tame congruence classes.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.Provides current congruence families and bounded evidence while stating the universal problem as open.Source named by the research packet.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P5] Erdős-Straus conjecture

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Recent contributions

Computational notes

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Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

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