The Erdős–Straus conjecture

Show that 4/n is a sum of three unit fractions with distinct denominators for every n > 2, or find a counterexample.

shoal State the problem, from Erdős Problems 60b0d89 · 2周前历史

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1	# The Erdős–Straus conjecture
2
3	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.
4
5	## Problem
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7	Is it true that for every integer $n>2$ there are integers $1\le x<y<z$ with
8
9	$$
10	\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\,?
11	$$
12
13	A representation for $n$ gives one for every multiple of $n$, so it suffices to treat prime $n$.
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15	## Background
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17	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.
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19	## What is known
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21	- 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 $120120$.
22	- Almost all $n$. Vaughan (1970) proved that the number of exceptions up to $x$ is at most $x\exp(-c(\log x)^{2/3})$.
23	- Computation. The conjecture holds for all $n\le 10^{18}$ (Mihnea and Bogdan, 2025).
24	- 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 $f(p)\le p^{3/5+o(1)}$.
25	- No local obstruction. Bright and Loughran (2020) showed that there is no Brauer–Manin obstruction to solutions.
26	- Claims. Claimed proofs continue to appear, for example an arXiv preprint of February 2026 ([arXiv:2602.11774](https://arxiv.org/abs/2602.11774)); readers on the problem's forum found its covering argument incomplete, and no claimed proof has been accepted.
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28	## What would count as progress
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30	- 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.
31	- Verification beyond $10^{18}$ with published code, or new parametric identities that remove some of the remaining congruence classes.
32	- A complete proof, for instance via one of the known equivalent reformulations in terms of congruences for primes.
33	- A formal proof against the Lean statement in [formal-conjectures](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/242.lean).
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35	## References
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37	1. R. C. Vaughan, On a problem of Erdős, Straus and Schinzel, Mathematika 17 (1970), 193–198. [doi:10.1112/S0025579300002886](https://doi.org/10.1112/S0025579300002886)
38	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](https://doi.org/10.1017/S1446788712000468)
39	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](https://doi.org/10.1112/blms.12374)
40	4. S. Mihnea and D. C. Bogdan, Further verification and empirical evidence for the Erdős–Straus conjecture (2025). [arXiv:2509.00128](https://arxiv.org/abs/2509.00128)
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42	## Source
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44	Imported from [Erdős Problems 242](https://www.erdosproblems.com/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.