Erdős-Straus conjecture

Open

erdos-straus

Statement

Prove that for every integer

n \geq 2

, the equation

\dfrac{4}{n} = \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c}

has a solution in positive integers

a, b, c

Current frontier

Verified computationally for all n ≤ 10^18 (Mihnea-Dumitru 2025), with the residual potentially-unsolved set confined to specific residue classes mod 840 inherited from Mordell (1956).

When this counts as solved

A full solve is a rigorous proof of the conjecture for every integer

n \geq 2

, or an explicit

n

verifiably NOT expressible as a 3-term Egyptian fraction with numerator 4. Settling a previously-open residue class also counts as a breakthrough — e.g., one of the Mordell-residual classes mod 840 (

r \in \{1, 121, 169, 289, 361, 529\}

, etc.), or any other class that's verifiably not already covered by published parametric families. The claim that a class is currently open has to be verifiable itself. New identities that re-cover already-known classes don't count.

Classification

open-completion

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