Open problem, Arithmetic and number theory, posed 1948

Erdős–Straus conjecture

OpenErdős

For every integer $n \ge 2$ , the rational number $4/n$ can be expressed as the sum of three positive unit fractions: $\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ for some positive integers $x, y, z$ .

Posed by, 1948Paul Erdős

Research frontier as of 2026

As of 2026, the Erdős–Straus conjecture remains open. Sieve methods (Vaughan 1970) show that the number of exceptions $n \le N$ is at most $O(N \exp(-c (\log N)^{2/3}))$ , so the conjecture holds for almost all integers in the sense of natural density. Modular filtering (Salez 2014 and subsequent computational extensions) has verified the conjecture up to at least $10^{17}$ . However, because the residue classes $p \equiv r^2 \pmod{q}$ cannot be covered by finitely many polynomial identities (Mordell's quadratic-residue obstruction), a full proof requires new analytic or arithmetic tools.

Best known results

Vaughan (1970): the number of integers $n \le N$ for which $4/n = 1/x + 1/y + 1/z$ has no solution is at most $O(N \exp(-c (\log N)^{2/3}))$ .
Elsholtz and Tao (2013): the average number of solutions over primes $p \le N$ is bounded between $c_1 \log^2 N$ and $c_2 \log^2 N \log \log N$ .
Salez (2014): computer verification confirms that no counterexample exists for $n \le 10^{17}$ .

Tools and where they stop

Tool	Achieved	Where it stops
Modular covering congruences and large sieves	Eliminates all residue classes except nonzero squares modulo any chosen modulus $M$ , proving density $0$ of exceptions and enabling verification up to $10^{17}$	Mordell's obstruction: $1$ is a quadratic residue modulo every integer $M$ , so no finite system of congruences can cover all primes
Parametric divisor-sum analysis on affine varieties (Elsholtz–Tao)	Establishes sharp upper and lower bounds on the average number of representations $\sum_{p \le N} f(p)$	Controls the average over primes $p \le N$ rather than guaranteeing $f(p) \ge 1$ for every individual prime $p$

Open questions

Does every prime $p \equiv 1 \pmod{24}$ admit a representation $4/p = 1/x + 1/y + 1/z$ in positive integers?
Does Schinzel's generalization hold — that for every fixed $m \ge 4$ , $m/n = 1/x + 1/y + 1/z$ is solvable for all sufficiently large $n$ ?

References

Christian Elsholtz, Terence Tao (2013). Counting the number of solutions to the Erdős–Straus equation on unit fractions · DOI:10.1017/S1446788713000244 · arXiv:1107.1010
L. J. Mordell (1969). Diophantine Equations
R. C. Vaughan (1970). On a problem of Erdős and Straus · DOI:10.1112/S0025579300002886
Serge E. Salez (2014). The Erdős–Straus conjecture: New modular equations and checking up to N = 10^17 · arXiv:1406.6307 [preprint, not peer-reviewed]