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Statement

The paper states the conjecture as: every fraction $4/n$ can be expanded as a sum of three unit fractions $1/x+1/y+1/z$ with $x,y,z\in\mathbb N^*$ (p. 1); no distinctness is required, and the case of prime $n$ suffices because a solution for $p$ scales to one for $kp$ (p. 1).

Section 2.1 (pp. 1--2), the reported result. "We improved this bound to $p\le10^{18}$ by extending this approach with $S_{29}$, obtaining a set $R_8$ with $|R_8|=2101514$ residue classes modulo $G_8=25878772920$ for which we must check the conjecture" (p. 2). Salez's modular filters $S_m$ (the residue classes modulo $m$ where the conjecture is known) give, from the first seven prime filters up to $S_{23}$, the set $R_7$ modulo $G_7$ behind Salez's verification to $10^{17}$ (p. 1); the paper adds $S_{29}$. The surviving integers are checked in batches $B_k=\{r+kG_8:r\in R_8\}$, up to $k=38641709$, against a precomputed set of $140000$ prime filters; the first $3864170$ batches are covered by the earlier $10^{17}$ result; the run took about two weeks (Sections 2.1--2.2, p. 2). The integers that no filter in the set removed were set aside, and the authors report that none of them is prime (Section 2.2, p. 2). The code is at github.com/esc-paper/erdos-straus (footnote 1, p. 2).

Source. Mihnea and Dumitru, arXiv:2509.00128v1 (29 August 2025), 4 pp.; Section 1 on p. 1, Section 2 from p. 1 to p. 2, read on the page images. No journal version was found (arXiv listing of 2026-09-18).

Read depth. Claims checked: the statements of Sections 1--3 were read clause by clause. This is a computation report with no theorem label: the result is the authors' statement that the computation completed, the computation was not rerun here, the code was not fetched, and the paper gives no independent check of its own run. The paper's second computation, the solution counts of Section 3, has its own page.

Dependencies

Salez's modular-filter algorithm and Salez's verification to $10^{17}$ (arXiv:1406.6307, 2014; not consulted); Mordell's residue classes modulo $840$.

Bears on

• Problem 242: the verification of all $n\le10^{18}$ that the site's commentary cites, recorded as a pending partial claim (Mihnea and Dumitru 2025); the paper checks primes, and composite $n\le10^{18}$ follow from their prime factors (p. 1). The paper's convention allows repeated denominators; a repeated-term solution converts into a distinct one with three terms (see the survey's Takenouchi remark on Theorem 1's page).