Conjecture Fieldbook

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Erdős–Straus conjecture

Canonical problem dossier

For every integer n ≥ 2, can 4/n be written as a sum of three positive unit fractions?

Status
Open
Field
Number theory
Verification
Reviewed dossier
Source grade
secondary
Reviewed
2026-07-27T18:04:33.834Z
Reviewer
Conjecture Fieldbook maintainers

Formal expression

4n=1x+1y+1z,x,y,z∈N\frac4n=\frac1x+\frac1y+\frac1z,\qquad x,y,z\in\mathbb{N}

Why it matters

It is an elementary-looking Diophantine problem that mixes congruences, constructive identities, and large-scale computation.

Context and known approaches

Many congruence classes admit direct identities, and enormous ranges have been checked, but a proof covering every n is unknown.

Formal boundary

For every n≥2, find positive integers x,y,z such that 4/n = 1/x + 1/y + 1/z.

Known partial results

Pending review. No structured partial-result note has been added to this dossier.

Source trail

  1. Wikipedia — History, identities, and computational evidence.
  2. Wikipedia open-problems list — Composite index by mathematical field.
  3. Wikipedia list subsection — Placement in the accepted revision-pinned living-list snapshot.