Problem 242

For every $n>2$ there exist distinct integers $1\leq x<y<z$ such that

\frac{4}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.

Source: erdosproblems.com/242

Number theory
Unit fractions

Wiki page

Status

ClaimedProved

A full solution has been claimed but not yet accepted. The statement is true.

Falsifiable on the site: the label is FALSIFIABLE (page last edited 7 May 2026), which the site explains as open but refutable by one finite counterexample. The standing derived from the claim pages is claimed, claim proved: three dated manuscripts claim the whole conjecture, Alomari's preprint of February 2023 (Alomari 2023), Dyachenko's arXiv preprint of 7 November 2025 (Dyachenko 2025) and Bradford's arXiv preprint of 12 February 2026 (Bradford 2026), none refereed, accepted by anyone or submitted to the site's proof-claim tab, and all pending. The one claim on the tab, Brian Akaka's AI-assisted lower bound of September 2026 on the number of solutions for almost all primes, settles the conjecture for no $n$ and has no claim page (see Forum and AI-assisted items). No proof and no counterexample was found in the search whose scope the Current assessment records. Two refereed partial results settle infinitely many $n$ and are accepted partial claims: Obláth's case where $n+1$ has a prime factor $\equiv3\pmod4$ (Obláth 1950) and Terzi's primes outside $198$ classes modulo $120120$ (Terzi 1971). The verification of all $n\le10^{18}$ is an unrefereed computation report, a pending partial claim (Mihnea and Dumitru 2025). Vaughan's bound on the exceptional set and the counting, equivalence and obstruction results settle no $n$ .

Er50c: Erdős, P., Az $1/x_1+\cdots+1/x_n=a/b$ egyenlet egész számú megoldásairól. Mat. Lapok 1 (1950), 192--210; the conjecture on printed p. 195, the English summary on p. 210. The site's source line carries the key; its reference list on the page omits it. Library home: Erdős 1950.
ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), p. 44 (the site gives no page). Library home: Erdős and Graham 1980.
Er61: Erdős, P., Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 221--254, and [Er79] Erdős, P., Some unconventional problems in number theory. Math. Mag. 52 (1979), 67--70: the site's source keys; not held.
Va99: Various, Some of Paul's favorite problems. Booklet produced for the conference "Paul Erdős and his mathematics", Budapest, July 1999 (1999), item 1.13, printed pp. 2--3: "(Erdős, Straus) Prove that for every $n>1$ $\frac4n=\frac1x+\frac1y+\frac1z$ is solvable in integers $x,y,z$ ", with $n>1$ and no distinctness printed. Library home: booklet card.
BlEl22: Bloom, Thomas F. and Elsholtz, Christian, Egyptian fractions. Nieuw Arch. Wiskd. (5) 23 (2022), no. 4, 237--245 (pages of the typeset article; also arXiv:2210.04496v1). Theorem 1, p. 239; Theorem 3 and the Vaughan sentence, p. 240. Library home: Bloom and Elsholtz 2022.
BrLo20: Bright, Martin and Loughran, Daniel, Brauer--Manin obstruction for Erdős--Straus surfaces. Bull. Lond. Math. Soc. 52 (2020), no. 4, 746--761, DOI 10.1112/blms.12374 (arXiv:1908.02526v2; preprint pages cited). Theorem 1.1, p. 1. Library home: Bright and Loughran 2020.
ElTa13: Elsholtz, Christian and Tao, Terence, Counting the number of solutions to the Erdős--Straus equation on unit fractions. J. Aust. Math. Soc. 94 (2013), no. 1, 50--105, DOI 10.1017/S1446788712000468 (arXiv:1107.1010v6; preprint pages cited). Theorem 1.1, p. 4; the corollary, p. 5; Proposition 1.7, pp. 6--7; Table 1, p. 4. Library home: Elsholtz and Tao 2013.
ElPl20: Elsholtz, Christian and Planitzer, Stefan, The number of solutions of the Erdős--Straus equation and sums of $k$ unit fractions. Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), no. 3, 1401--1427, DOI 10.1017/prm.2018.137 (arXiv:1805.02945v1; preprint pages cited). Theorem 1 and Corollary 1, p. 2; Theorem 3, pp. 3--4. Library home: Elsholtz and Planitzer 2020.
MiDu25: Mihnea, S. and Dumitru, B. C., Further verification and empirical evidence for the Erdős--Straus conjecture. arXiv:2509.00128v1 (29 August 2025), 4 pp.; a computation report, not refereed. Section 2, p. 2. Library home: Mihnea and Bogdan 2025.
PoWe25: Pomerance, C. and Weingartner, A., Exceptions to the Erdős--Straus--Schinzel conjecture. The Ramanujan Journal 69 (2026), no. 2, article 31, DOI 10.1007/s11139-025-01312-2 (arXiv:2511.16817v2; preprint pages cited). Theorems 1.1--1.3, p. 2. Library home: Pomerance and Weingartner 2025.
Mo69: Mordell, L. J., Diophantine equations. Academic Press (1969), Ch. 30 per [BrLo20]. Not held; the six residue classes modulo $840$ are quoted second-hand.
Ob50: Obláth, R., Sur l'équation diophantienne $\frac4n=\frac1{x_1}+\frac1{x_2}+\frac1{x_3}$ . Mathesis 59 (1950), 308--316. Not held; no open archive was found.
Si56: Sierpiński, W., Sur les décompositions de nombres rationnels en fractions primaires. Mathesis 65 (1956), 16--32. Not held.
Te71: Terzi, D. G., On a conjecture by Erdős--Straus. Nordisk Tidskr. Informationsbehandling (BIT) 11 (1971), 212--216, DOI 10.1007/BF01934370 (received 23 September 1970). Rosati's conditions (2)--(3), p. 212; the six classes modulo $840$ and Table 1, p. 213; congruence (9) and Table 2, p. 214; the verification statement and Table 3, p. 215. Library home: Terzi 1971.
Va70: Vaughan, R. C., On a problem of Erdős, Straus and Schinzel. Mathematika 17 (1970), 193--198, DOI 10.1112/S0025579300002886 (received 6 February 1970). The definition of $E_a(N)$ and the Theorem, p. 193; the sieve inequality (5), p. 194; the closing estimate, p. 198. [PoWe25], Theorem 1.3, restates the bound with an explicit dependence on the numerator. Library home: Vaughan 1970.

Problem 242

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