The Erdős-Straus Conjecture

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

A unit fraction is a rational number $1/u$ with $u\in{\mathbb{N}}$. For each integer $n\geq2$, seek $x,y,z\in{\mathbb{N}}$ satisfying $4xyz=n(xy+xz+yz)$. This integer equation is equivalent to the displayed unit-fraction equation. The three denominators may coincide; requiring them to be distinct would change the statement. [S1] [S2]

Statement recorded in the supplied source.

For every integer $n \geq 2$, the equation $\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ has a solution in positive integers x, y, z. [S1]

Short English statement

Can four divided by any integer greater than one always be expressed as the sum of three positive unit fractions?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 60 (NT-012), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 The Erdős-Straus Conjecture and the Structure of..., INTEGERS 26 (2026). https://math.colgate.edu/~integers/aa42/aa42.pdf. Bibliographic information from the supplied record.