The Erdős–Straus conjecture

Can $4/n$ always be written as a sum of three unit fractions with distinct denominators? The answer is yes for every $n$ up to $10^{18}$ and for almost all $n$, but no proof covers every $n$. A single counterexample, which would have to be a large prime, would settle the question the other way.

Problem

Is it true that for every integer $n>2$ there are integers $1\le x<y<z$ with

$$\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\,?$$

A representation for $n$ gives one for every multiple of $n$, so it suffices to treat prime $n$.

Background

The conjecture is due to Erdős and Straus; perhaps its first appearance in print is a paper of Obláth (1950, submitted 1948) that describes it as a conjecture of Erdős. A greedy algorithm easily writes $4/n$ as a sum of at most four distinct unit fractions, so the question is whether three always suffice. Schinzel proposed a generalisation in which $4$ is replaced by any fixed numerator $a$ and $n$ is large in terms of $a$; for $a=5$ this is due to Sierpiński. The conjecture is notorious for attracting flawed proofs, and claimed solutions should be treated with caution until they are refereed.

What is known

• Congruence classes. Using parametric families of solutions, Mordell proved the conjecture for all $n$ except possibly those congruent to $1,121,169,289,361$ or $529$ modulo $840$; Terzi (1971) reduced the possible exceptions to $198$ classes modulo $120120$.
• Almost all $n$. Vaughan (1970) proved that the number of exceptions up to $x$ is at most $x\exp(-c(\log x)^{2/3})$.
• Computation. The conjecture holds for all $n\le 10^{18}$ (Mihnea and Bogdan, 2025).
• Counting solutions. Elsholtz and Tao (2013) showed that the number $f(p)$ of solutions satisfies $\sum_{p\le N}f(p)=N(\log N)^{2+o(1)}$ and $f(p)\le p^{3/5+o(1)}$.
• No local obstruction. Bright and Loughran (2020) showed that there is no Brauer–Manin obstruction to solutions.
• Claims. Claimed proofs continue to appear, for example an arXiv preprint of February 2026 (arXiv:2602.11774); readers on the problem's forum found its covering argument incomplete, and no claimed proof has been accepted.

What would count as progress

• A counterexample: a prime $p$ for which a computer search, made exhaustive by bounding $x$, $y$ and $z$, finds no representation; this is checkable by rerunning the search.
• Verification beyond $10^{18}$ with published code, or new parametric identities that remove some of the remaining congruence classes.
• A complete proof, for instance via one of the known equivalent reformulations in terms of congruences for primes.
• A formal proof against the Lean statement in formal-conjectures.

References

1. R. C. Vaughan, On a problem of Erdős, Straus and Schinzel, Mathematika 17 (1970), 193–198. doi:10.1112/S0025579300002886
2. C. Elsholtz and T. Tao, Counting the number of solutions to the Erdős–Straus equation on unit fractions, J. Aust. Math. Soc. 94 (2013), 50–105. doi:10.1017/S1446788712000468
3. M. Bright and D. Loughran, Brauer–Manin obstruction for Erdős–Straus surfaces, Bull. London Math. Soc. 52 (2020), 746–761. doi:10.1112/blms.12374
4. S. Mihnea and D. C. Bogdan, Further verification and empirical evidence for the Erdős–Straus conjecture (2025). arXiv:2509.00128

Source

Imported from Erdős Problems 242, curated by Thomas Bloom. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.