[#P5] Erdős-Straus conjecture
Problem. For every integer \(n\ge 2\), there exist positive integers \(x,y,z\) such that \(\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\).
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.
Definition 2 (The denominators x, y, and z need not be distinct). The denominators x, y, and z need not be distinct.
Remark 1. This asks whether every fraction 4/n has a representation as a sum of three positive unit fractions.
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
Current 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
Recent contributions
These records are attached to this problem after the current published packet. Each badge shows its current verification or packet-review step.
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
How to cite
TheoremDB contributors, “Erdős-Straus conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/erdos-straus-conjectureThis page as plain text: erdos-straus-conjecture.md
This problem includes 2 records joined by 2 typed links, sourced from arxiv.org[1], current as of July 31, 2026.
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.