Erdös–Straus conjecture (m = 4)

Prove that for every integer n ≥ 2 there exist integers x, y, and z such that 4/n = 1/x + 1/y + 1/z.

Background

The paper studies polynomial solutions to the Diophantine equation m/n = 1/x + 1/y + 1/z in residue classes and discusses restrictions when m does not divide the modulus n1. As context, it recalls the classical Erdös–Straus conjecture for m = 4.

Despite extensive partial results and computational verifications reported in the literature, the conjecture has not been proven. The authors cite it to situate their results within the broader landscape of Egyptian fraction problems.

References

Two well known conjectures by Erdös-Straus and Sierpinski state that the diophantine equation m/ n =1/ x+1/ y+1/ z has integer solutions x, y,zfor every integer n ≥ 2and m = 4 (Erdös-Straus) or m = 5 (Sierpinski). There is an impressive body of evidence for the validity of both conjectures, see e.g. [1,2,3,5], but no valid proof.

— Unique polynomial solution of $m/n=1/x+1/y+1/z$ for $n \equiv b {\rm mod}\, a$ if $(a,m)=1$  (2404.01307 - Schuh, 2024) in Section I (Introduction)

A well known conjecture by Erdos-Straus (ESC in the following) states that the Diophantine equation 4/n-(1/x+1/y+1/z)=0 (1) is solvable in every integer n ≥2.

— The Erdös-Straus Conjecture and Pythagorean Primes  (2503.11672 - Schuh, 26 Feb 2025) in Section I Introduction (page 1)

The $\tau_3$-type bound for arbitrary witness systems. Proposition~\ref{prop:optimality_23} pins the exponent $2/3$ for the two witness systems exhibited here. Whether every system attached to the Type~I/Type~II dichotomy satisfies $\sum_{p\le Z}\varrho(p)/p\ll(\log Z){2}$, which is what a genuine impossibility statement for this method would require, is not proved.

— Sieve dimension and search depth for the Erdős-Straus conjecture, $n \equiv 1 \pmod{24}$  (2608.24035 - Dahan, 25 Aug 2026) in Section 8, “Conclusion and open questions”; Proposition 5.19 and Section 5.2.7

A sub-quadratic factorization-free interval procedure. Theorem~\ref{thm:batch_complete} locates the whole quadratic cost of a complete decision on $[N,2N]$ in the extraction of the divisors of $4u{2}d+1$ in the Type~I pass. Whether a sub-quadratic factorization-free procedure exists by some other route is left open.

— Sieve dimension and search depth for the Erdős-Straus conjecture, $n \equiv 1 \pmod{24}$  (2608.24035 - Dahan, 25 Aug 2026) in Section 8, “Conclusion and open questions”; Theorem 4.14 and Section 4.5

Whether $\bigcap_J E_2(N;J)$ is empty. By Theorem~\ref{thm:gen_criterion} this is exactly the question of whether every prime $n\equiv1\pmod{24}$ has a Type~II solution. Theorem~\ref{thm:fixed_depth} shows that the intersection has counting function $\ll_A N(\log N){-A}$ for every $A$, and Corollary~\ref{cor:H_density} gives the same for the one-parameter analogue $\mathcal{H}$, whose fourteen known members all disappear at depth $5$ once the modulator is free; but emptiness is a different matter, and the general conjecture remains open.

— Sieve dimension and search depth for the Erdős-Straus conjecture, $n \equiv 1 \pmod{24}$  (2608.24035 - Dahan, 25 Aug 2026) in Section 8, “Conclusion and open questions”; Theorem 5.11 and Definition 5.1