Problem 242

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claims/: The 6 claim pages of Problem 242, one per claimant's result; the problem's standing derives from them.

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

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

Formulation. The site's wording of 2026-09-18 (page last edited 7 May 2026). This is the Erdős--Straus conjecture. The literature usually states it for positive integers $x,y,z$ that need not be distinct and for $n\ge2$; the two forms agree for $n>2$. The survey [BlEl22] (p. 237) records Takenouchi's observation that a sum of $k$ unit fractions with repeated terms is also a sum of $k$ distinct unit fractions: a repeated term is replaced through $\frac1{2t}+\frac1{2t}=\frac1{t+1}+\frac1{t(t+1)}$ or $\frac1{2t+1}+\frac1{2t+1}=\frac1{t+1}+\frac1{(t+1)(2t+1)}$, which raises the sum of the denominators, until no term repeats; a representation with fewer than three terms is first split by $1/y=1/(y+1)+1/(y(y+1))$. The one sum the substitutions leave fixed, $\frac12+\frac12=1$, is the case $n=4$, where $1=\frac12+\frac13+\frac16$. The case $n=2$ is excluded because $4/2=2$ exceeds $1+1/2+1/3$. It suffices to prove the statement for prime $n$, since a solution for $p$ scales to one for every multiple of $p$ (site commentary; [ElTa13], p. 3). The earliest statement in the library is Erdős's 1950 paper (p. 195): together with Straus he conjectures that $4/b$ is a sum of at most three distinct unit fractions for every $b>4$, and Straus had verified this for $4<b<5000$.

Status. 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 $n$.

Source. erdosproblems.com/242, accessed 2026-09-18: the problem page (FALSIFIABLE, explained by the site as open but refutable by a finite counterexample; last edited 7 May 2026; source keys [Er50c], [Er61], [Er79], [ErGr80], [Va99, 1.13]; additional thanks to Alfaiz and Bryce Orloski), its discussion thread (18 comments shown, 9 August 2025 to 13 February 2026; four further replies under a "Show 4 more comments" control are not recorded here) and its proof-claim tab with one partial claim (submitted 16 September 2026). Cite as: T. F. Bloom, Erdős Problem #242, https://www.erdosproblems.com/242, accessed 2026-09-18.

References.

• [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: erdos_1950_az_egyenlet_egesz_szamu_megoldasairol_diophantine.
• [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: erdos_1980_old_new_problems_results_combinatorial_number_theory.
• [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_2022_egyptian_fractions.
• [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_2020_brauer_manin_obstruction_erdos_straus.
• [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_2013_counting_number_solutions_erdos_straus.
• [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_2020_number_solutions_erdos_straus_equation.
• [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_2025_further_verification_empirical_evidence_erdos_straus.
• [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_2025_exceptions_erdos_straus_schinzel.
• [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_conjecture_erdos_straus.
• [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_erdos_straus_schinzel.

Formalization. Statement only. The file ErdosProblems/242.lean of formal-conjectures at the linked revision (main) declares erdos_242 (n : ℕ) (hn : 2 < n) : ∃ x y z : ℕ, 1 ≤ x ∧ x < y ∧ y < z ∧ (4 / n : ℚ) = 1 / x + 1 / y + 1 / z under category research open with proof sorry, and the variant erdos_242.variants.schinzel_generalization (for each $a>0$ and all sufficiently large $n$, the same with $a/n$; research open, sorry). The community database (problems.yaml) records the problem as falsifiable (last update 28 September 2025), the statement as formalized since 31 August 2025, no formal proof, and the OEIS entries A073101, A075245--A075248 and A287116. The file was not built.

Current assessment

The question (site formulation of 2026-09-18). The statement above; FALSIFIABLE; last edited 7 May 2026. The commentary names the conjecture and traces its first appearance in print to Obláth's paper [Ob50], submitted in 1948, where it is attributed to Erdős; notes that the greedy algorithm gives a representation of $4/n$ by at most four distinct unit fractions; states Schinzel's generalization (for fixed $a$ and all large $n$, $a/n$ is a sum of three distinct unit fractions; Sierpiński's conjecture for $a=5$; [PoWe25] for background); says it suffices to treat prime $n$ and cites [MiDu25] for the verification of all $n\le10^{18}$; and lists partial results: Obláth (true when $n+1$ has a prime factor $\equiv3\pmod4$, hence for almost all $n$), Mordell (true for all $n$ not congruent to one of $\{1,121,169,289,361,529\}$ modulo $840$), Terzi (all $n$ outside $198$ bad classes modulo $120120$), Vaughan (the number of exceptions in $[1,x]$ is $\le x\exp(-c(\log x)^{2/3})$), the equivalence with a covering of the primes by congruence classes (Theorem 1 of [BlEl22]), Bright and Loughran's absence of a Brauer--Manin obstruction, Elsholtz and Tao's $\sum_{p\le N}f(p)=N(\log N)^{2+o(1)}$ and $f(p)\le p^{3/5+o(1)}$, and Elsholtz and Planitzer's $f(n)\ge(\log n)^{\log6+o(1)}$ for almost all $n$. The proof-claim counter shows one claim; the thread has 18 comments. The community database record (above) agrees with the label. The label records the question's logical form, not its state of knowledge: a counterexample is a single $n$ for which the finitely many candidate triples (each $x\le3n/4$, and $y,z$ bounded once $x$ is fixed) all fail, a check by finite arithmetic, while a proof must cover all $n$.

Origin. Erdős 1950, printed p. 195 (result page): with $N(a,b)$ the least number of distinct unit fractions summing to $a/b$, Erdős writes (in Hungarian; rendered, not quoted) that he and Straus conjecture $N(4,b)\le3$ for $b>4$ and that Straus proved this for $4<b<5000$; the English summary (p. 210) writes $N(4,b)<4$ for every $b\ge4$. The 1980 monograph repeats it on printed p. 44: "An old conjecture of Erdös and Straus asserts that for all $n>1$, the equation $(*)$ $\frac4n=\frac1x+\frac1y+\frac1z$ has integer solutions. This has still not been settled." The page then credits Vaughan [Va (70)] and Webb [Web (70)] with estimates for the number $f(N)$ of $n\le N$ without a solution, from which $f(N)<N\exp\{-c(\log N)^{2/3}\}$ for some $c>0$, records the verification of $(*)$ for $n\le10^8$ (citing [Franc (78)], [Ter (71)], [Ya (64)] and [Ya (65)]), and continues with the Schinzel--Sierpiński generalization and Schinzel's $\pm$ variant, "proved for all $a\le40$". The survey [BlEl22] (p. 239) adds that Obláth's paper attributes the conjecture to Erdős and that Erdős, asked in 1996, said $4$ is "the first interesting case".

What is proved (recorded at statement level).

• The covering-congruence form. Theorem 1 of the survey (p. 239): the conjecture holds if and only if every prime lies in a class $-a/c\pmod{4acd-1}$ for some $a,c,d\ge1$ or a class $-(4c^2d+1)/k\pmod{4cd}$ for some $c,d,k\ge1$ with $k\mid4c^2d+1$. The one-page proof (sufficiency by two explicit identities, necessity by a gcd argument) is recorded in outline only, not verified here. The survey also lists the classes modulo $840$ not covered by the simplest identities as $1,49,121,169,289,361$ (p. 239), where the site, [MiDu25] (p. 2) and the thread's quotation of Mordell give $1,121,169,289,361,529$; $49$ is a multiple of $7$ and $4/(7k)=1/(2k)+1/(14k)$, so the survey's list has a misprint; Mordell's own list ([Mo69], not held) remains second-hand, and [Te71] (p. 213) prints the same six classes $1,121,169,289,361,529\pmod{840}$ as "the result of K. Yamomoto [sic]", recovered by its first algorithm at $M=840$.
• Finite verification. [MiDu25], Section 2 (p. 2): a modular-filter computation extending Salez's verification for primes up to $10^{17}$ (arXiv:1406.6307, 2014; cited from its abstract, paper not held) to all primes $p\le10^{18}$, through $2101514$ residue classes modulo $25878772920$ and $140000$ prime filters, in about two weeks; composite $n\le10^{18}$ follow from their prime factors. This is the authors' report of a completed computation, not rerun by the corpus, and a pending partial claim, Mihnea and Dumitru 2025. Table 1 of [ElTa13] (p. 4) gives the earlier history: Straus $5000$ (by 1950), Bernstein $8000$ (1962), Shapiro $20000$, Obláth $106128$ (1948/9), Rosati $141648$ (1954), Yamamoto $10^7$ (1964), Jollensten $1.1\times10^7$ (1976), Terzi $10^8$ (1971), Elsholtz and Roth $10^9$ to $1.6\times10^{11}$ (unpublished, 1994--96), Kotsireas $10^{10}$ (1999), Swett $10^{14}$ (1999), Bello-Hernández, Benito and Fernández $2\times10^{14}$ (2012), Salez $10^{17}$ (2014), with the caveats that Terzi's set of checked primes appears incomplete and that Franceschine's $10^8$ is not an independent verification. Terzi's own statement is first-hand ([Te71], p. 215): "With the help of the second algorithm the correctness of the Erdös--Straus conjecture is now proved for all $n\le10^8$", with Obláth credited for $n<106129$, Rosati for $106129\le n<141649$, Yamamoto for $n\le10^7$ and the paper's own run on a BESM-6 for $10^7<n\le10^8$. Its Table 3 prints Rosati quadruples $(a,b,c,d)$ for seven primes, introduced as the solutions for all primes of its $198$ classes in that interval, while there are $43485$ such primes (a sieve count), so the printed record supports [ElTa13]'s caveat; two rows are misprinted (the row for $34954921$ satisfies the paper's identity with $a=118091$ for the printed $1118091$, and the row for $43950481$ with its $c$ and $d$ exchanged, $c=39$, $d=453$). The computation is the author's report and was not rerun.
• The exceptional set. Vaughan's bound is first-hand ([Va70], the Theorem, p. 193): with $E_a(N)$ the number of $n\le N$ for which $a/n=1/x+1/y+1/z$ has no solution in positive integers, for each fixed positive integer $a$, $E_a(N)\ll N\exp\{-(\log N)^{2/3}/C(a)\}$ with $C(a)>0$ depending at most on $a$; $a=4$ is the Erdős--Straus case, the form $N\exp(-c(\log N)^{2/3})$ quoted by the survey (p. 240) and the monograph (p. 44), and the paper notes that almost every $n$ is therefore representable. The proof (pp. 193--198) is recorded in outline only, not verified here: explicit solutions from the congruence $rn+s\equiv0\pmod{arst-1}$, Montgomery's large sieve, the Bombieri--Vinogradov theorem for the average of the sifted class counts, and Rankin's method. The uniform form is Theorem 1.3 of Pomerance and Weingartner (p. 2): for $4\le m\le\log^2N$ the number of $n\le N$ with $m/n$ not a sum of three unit fractions is at most $N/\exp(C\log^{2/3}(N)/\varphi(m)^{1/3})$, with $m=4$ the Erdős--Straus case. Elsholtz's generalization of Vaughan's bound to $m/n$ as a sum of $k$ unit fractions is recorded as the survey's restatement (Theorem 3, p. 240). Obláth's almost-all result and Mordell's classes are second-hand: the site's commentary, [PoWe25] pp. 1--2 ("An early result of Obláth [11] is that $n$ has this property if $n+1$ is divisible by a prime $p\equiv3\pmod4$. This implies that asymptotically all $n$ have the Erdős--Straus property") and [ElTa13] p. 4. Obláth's result appeared in a journal and is an accepted partial claim, Obláth 1950. Mordell's statement, every $n$ outside the six classes modulo $840$, is in a book and has no claim page, because it follows from Terzi's result for primes: the six classes are exactly the squares of the units modulo $840$, which form a group under multiplication, so an $n$ coprime to $840$ outside them has a prime factor outside them, and that prime is $11$, $13$ or a prime outside Terzi's $198$ classes, which refine the six; an $n$ not coprime to $840$ has one of $2,3,5,7$ as a factor; and for $p=2,3,5,7,11,13$ the fraction $4/p$ is a sum of three unit fractions directly, repetition allowed. Terzi's classes are first-hand ([Te71], Table 2, p. 214): congruence (9), "$n\equiv N_2\pmod{120120}$ where $N_2$ takes all 198 values from Table 2", is the condition his first algorithm leaves as the only one under which, for a prime $n$, "the Erdös--Straus conjecture may happen to be untrue" (p. 213); every prime coprime to $120120$ outside the $198$ classes satisfies Rosati's parametrization and has a representation, with repetition allowed in the paper's convention. The classes refine Table 1's $34$ classes modulo $9240$ and the six classes modulo $840$; the $198$ values are transcribed as printed on the result page, their consistency with the coarser tables was checked, and the algorithm producing them is recorded in outline only. The result for primes, with every multiple of such a prime, is an accepted partial claim, Terzi 1971; the classes are not closed under multiplication, so a composite $n$ outside them may have all its prime factors inside them, and the site's reading, every $n$ outside the classes, says more than the paper.
• Counting solutions. With $f(n)$ the number of positive triples (repetition and order allowed), Theorem 1.1 of Elsholtz and Tao (p. 4) gives $N\log^3N\ll\sum_{n\le N}f_{\mathrm I}(n), \sum_{n\le N}f_{\mathrm{II}}(n)\ll N\log^3N$ and the prime sums of order $N\log^2N$ (the Type I upper bound with a factor $\log\log N$), whence $N\log^2N\ll\sum_{p\le N}f(p)\ll N\log^2N\log\log N$ (p. 5); their Proposition 1.7 (pp. 6--7) gives $f(p)\ll p^{3/5+O(1/\log\log p)}$ for every prime $p$; Elsholtz and Planitzer's Theorem 1 and Corollary 1 (p. 2) extend the upper bound to $O_\varepsilon(n^{3/5+\varepsilon})$ for every $n$, and their Theorem 3 (pp. 3--4) gives $f_3(4,n)\ge\exp((\log6+o(1))\log\log n) =(\log n)^{\log6+o(1)}$ on a set of $n$ of density one and $\exp((\log6+o(1))\log n/\log\log n)$ for infinitely many $n$. An average or a density-one lower bound does not give $f(n)>0$ for every $n$; the papers say so themselves (Remark 1.2 of [ElTa13] calls the Poisson heuristic drawn from these counts "only a heuristic").
• Obstructions. Theorem 1.1 of Bright and Loughran (p. 1): for every $n\ge2$ there is no Brauer--Manin obstruction to natural-number solutions on the surface $4u_1u_2u_3=n(u_1u_2+u_1u_3+u_2u_3)$; their Theorem 1.2 gives the necessary condition $\prod_{p\mid n}(-u_1/u_3,-u_2/u_3)_p=-1$ for odd $n$. This closes one route to a disproof and proves no existence.
• Schinzel's generalization. Theorem 1.1 of Pomerance and Weingartner (p. 2): any threshold $n_m$ beyond which $m/n$ is always a sum of three unit fractions exceeds $\exp(m^{1/3-\varepsilon})$ for large $m$; this concerns varying $m$ and says nothing about $m=4$.

The counting papers count positive triples without distinctness ([ElTa13] p. 2; [ElPl20] nondecreasing tuples; [PoWe25] display (2.1); [MiDu25] p. 1); existence is unaffected (Formulation), and the counts differ from the distinct-ordered count by bounded factors. OEIS A073101 counts solutions with $0<x<y<z$ and begins $0,0,1,1,2,5,5,6,4,9$, consistent with the statement's $n>2$.

Forum and AI-assisted items (leads with provenance, not status). The site states on the page that it does not verify the comments or claims that appear there.

• Proof-claim tab: one partial claim, Brian Akaka's AI-assisted lower bound of at least $c(\log p)^3$ Type II solutions (solutions $x<y<z$ with $p\mid y$, $p\mid z$ and $p\nmid x$, the convention of Elsholtz and Tao) for all but at most $CN/(\log N)^4$ primes $p\le N$, for every $N$ beyond a threshold, posted on Zenodo on 14 September 2026 as version v0.7 of the working preprint A cubic lower bound for strict Erdős--Straus solution counts at almost all primes (record 22754703, CC BY 4.0, with a source archive that the record's description says carries a Lean verification resting on external Bombieri--Vinogradov modules) and submitted to the tab on 16 September 2026, with the Zenodo record given as both proof and formalization link. The tab names the AI systems ChatGPT (Astra, Sol), Claude (Fable, Opus) and Kimi (K3). The claimant's description of the method: a restricted family of solutions with parameters $M=3abu$ and $s=3a^2u$ under the congruence $4M-1\mid p+4s$, counted without double counting; an average over residue classes of order $(\log B)^3$ against a variance of the same order; the Bombieri--Vinogradov theorem to pass from the residue model to the primes; and a squared-error bound for the primes with too few solutions. Earlier versions of 6 and 12 September 2026 proposed the weaker bound $c(\log p)^3/\log\log p$ outside an exceptional set of order $N(\log\log N)^2/(\log N)^3$; version v0.8 of 22 September 2026 extends the result to numerators $4\le m\le(\log X)^3$ and says that it does not prove the Erdős--Straus or Schinzel conjecture. No comment stands under the claim and no one has accepted it. It has no claim page: a counting lower bound outside an exceptional set of $CN/(\log N)^4$ primes settles the conjecture for no $n$, since an exceptional prime may have no solution at all, so it is not a claim about the problem; it is a result of the kind of the counting theorems of Elsholtz and Tao and of Elsholtz and Planitzer above, and like them it is recorded as a lead.
• Discussion, 13 February 2026: a comment reports a preprint by K. Bradford claiming a solution (arXiv:2602.11774v1, 12 February 2026, "A solution to the Straus-Erdős conjecture"; its abstract says the paper "outlines a solution" with positive $x\le y\le z$ for each prime $p$); a second commenter reads the preprint's final sentence on the covering system as a sign that the argument is incomplete and notes that none of Mordell's six residues is excluded; a third advises giving no attention to new preprints on this problem without publication, an author track record, realistic partial claims, an expert vouching or a proper formalization, and links a chat transcript, which is not a source. The preprint is a dated manuscript claiming the whole conjecture, so it has its own claim page, Bradford 2026, pending: it is consumed at the level of its abstract and arXiv record, no record refutes it, and no one has accepted it.
• Discussion, 27--29 January 2026: a commenter announces a Lean development (repository leochlon/erdstrau) claiming sorry-free proofs for several residue classes modulo $420$ and $840$ and a reduction of the conjecture to one construction; another commenter links its file for the class $529\pmod{840}$ and the site's owner concludes that the file proves nothing, being a finite check with an appeal to periodicity, while noting that a genuine formalization of the known congruence cases would be valuable.
• Discussion, 18--24 November and 7 December 2025 (account Alfaiz): a list of historical verification ranges partly at variance with Table 1 of [ElTa13] (Rosati $171649$ against the table's $141648$); M. W. Alomari's preprint A simple direct proof of the Erdős–Straus conjecture (February 2023, on Authorea, OSF and Research Square), which claims the whole conjecture and has its own claim page, Alomari 2023, and which a reply calls very mistaken; B. Ghermoul's arXiv:2508.07367 (10 August 2025), which claims an almost complete proof of Sierpiński's conjecture on $5/a$, not this problem; the commenter strongly doubts both, and the site's owner adds that many purported proofs of the conjecture have appeared over the years, none of which the site's owner found credible; Li Delang's bound $cN/(\log N)^k$ for the number of exceptions (a 1981 J. Number Theory paper, not held); the Vaughan and Pomerance--Weingartner papers; Mordell's residues as squares. Discussion, 1 February 2026: the site's owner notes that the case of almost all prime denominators follows from Vaughan's result.
• arXiv leads (abstracts only, API search for "Erdős--Straus" in abstracts, 30 records): 2026 preprints on the conjecture by Jiang (2609.09204, counting Type I and II solutions, "We do not address the Erdos-Straus conjecture itself"; its arXiv listing marks the paper withdrawn), Dahan (2608.24035, sieve dimension and search depth for $n\equiv1\pmod{24}$), Bello-Hernández, Benito and Fernández (2606.10922, a divisor parametrization), Ventas (2605.04551, heuristic finiteness of counterexamples) and Mballa (2602.20036, 23 February 2026, explicit solutions), and three 2025 preprints: Mballa's 2502.20935 (28 February 2025, revised 16 February 2026; a "partial resolution") and two by Dyachenko: 2511.07465 (7 November 2025), whose abstract claims a representation $4/P=1/A+1/(bP)+1/(cP)$ for every prime $P\equiv1\pmod4$, the whole conjecture once the classical case $P\equiv3\pmod4$ and the reduction to primes are added, and which therefore has its own claim page, Dyachenko 2025, pending; and 2511.17716, on $5/P$, which is not this problem. None of these listings carries a journal reference or acceptance evidence. Neither Mballa preprint has a claim page, because neither claims an $n$ beyond the classical cases: 2502.20935 gives explicit formulas that verify the conjecture only under a divisibility condition or a perfect-square condition that it conjectures and does not prove, and 2602.20036 gives explicit solutions for $n\equiv0,2,3\pmod4$ and for $n\equiv1\pmod4$ with a divisor $b\equiv3\pmod4$, all of which already have a prime factor $\equiv3\pmod4$ or are even and so fall under the classical identities, and its density-one statement is weaker than Vaughan's bound.
• OEIS: A073101 (solutions with $0<x<y<z$), A075245--A075247 (the solution with the largest $z$), A075248 (the count for $5/n$), A287116 (nonsquare integers not of the form $4M-d$ with $ab\mid M$ and $d\mid a+b$).

Search scope. The status rests on these routes; none found a proof, a counterexample or an accepted resolution.

• The site: problem page, discussion thread, proof-claim tab; the community database record; formal-conjectures 242.lean at the pinned commit.
• The primary sources, at statement level: [Er50c] (pp. 195 and 210), [ErGr80] (p. 44), [BlEl22] (pp. 239--240), [BrLo20] (pp. 1--3), [ElTa13] (pp. 2--7), [ElPl20] (pp. 2--4), [MiDu25] (pp. 1--3), [PoWe25] (pp. 2--4).
• Publication records: arXiv listings of 2210.04496, 1908.02526, 1107.1010, 1805.02945, 2509.00128, 2511.16817, 1406.6307, 2602.11774 and 2608.24035 (versions and journal references); Crossref records for [BrLo20], [ElTa13], [ElPl20] and [PoWe25] (the Ramanujan Journal record); Semantic Scholar for [PoWe25] (no record of the Bradford preprint was obtained).
• arXiv API metadata search for "Erdős--Straus" in abstracts (30 records, the 2026 ones listed above); the GitHub API for the thread's repository (404); the Zenodo API for record 22754703; OEIS for the six entries.

Not searched: MathSciNet, zbMATH, Google Scholar full text, X, ResearchGate. Not held: [Mo69], [Ob50], [Si56] (Mathesis; no archive found), Webb (1970), Li Delang (1981), Salez (2014, arXiv only), [Er61], [Er79]. [Te71] and [Va70] lie outside this search and are cited above from their library cards.

Remaining gaps. (1) The classical partial results of Obláth and Mordell are quoted second-hand, Obláth's an accepted partial claim on its refereed publication and Mordell's a consequence of Terzi's; Vaughan's bound is first-hand ([Va70], p. 193; its proof is recorded in outline only), with [PoWe25]'s Theorem 1.3 as its uniform form; the survey's Mordell list carries a misprint. Terzi's $198$ classes are first-hand and an accepted partial claim for primes, but his $10^8$ verification is an author's report whose printed Table 3 covers seven of the $43485$ primes it is said to cover and carries two misprinted rows. (2) The $10^{18}$ verification is a computation report that was not rerun. (3) The three pending full claims, Alomari 2023, Dyachenko 2025 and Bradford 2026, are unrefereed manuscripts consumed at the level of their abstracts and accepted by no one; the lower-bound claim on the tab settles no $n$. (4) The 1961 and 1979 Erdős sources are not held. There is no accepted proof to compile.

Proof coverage. No accepted proof exists; the standing claimed rests on three pending manuscripts. The partial results are recorded at statement level on their result pages; the survey's equivalence proof is recorded in outline only; no proof has been rewritten or independently reviewed.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1980_old_new_problems_results_combinatorial_number_theory
• various_1999_some_pauls_favorite_problems
• bloom_2022_egyptian_fractions
• bloom_2022_egyptian_fractions / theorem_1
• bloom_2022_egyptian_fractions / theorem_3
• bright_2020_brauer_manin_obstruction_erdos_straus
• bright_2020_brauer_manin_obstruction_erdos_straus / corollary_1_3
• bright_2020_brauer_manin_obstruction_erdos_straus / corollary_1_4
• bright_2020_brauer_manin_obstruction_erdos_straus / theorem_1_1
• bright_2020_brauer_manin_obstruction_erdos_straus / theorem_1_2
• bright_2020_brauer_manin_obstruction_erdos_straus / theorem_1_5
• bright_2020_brauer_manin_obstruction_erdos_straus / theorem_1_6
• bright_2020_brauer_manin_obstruction_erdos_straus / theorem_1_8
• bright_2020_brauer_manin_obstruction_erdos_straus / theorem_1_9
• elsholtz_2013_counting_number_solutions_erdos_straus
• elsholtz_2013_counting_number_solutions_erdos_straus / proposition_1_4
• elsholtz_2013_counting_number_solutions_erdos_straus / proposition_1_6
• elsholtz_2013_counting_number_solutions_erdos_straus / proposition_1_7
• elsholtz_2013_counting_number_solutions_erdos_straus / proposition_1_9
• elsholtz_2013_counting_number_solutions_erdos_straus / theorem_1_1
• elsholtz_2013_counting_number_solutions_erdos_straus / theorem_1_11
• elsholtz_2013_counting_number_solutions_erdos_straus / theorem_1_8
• elsholtz_2020_number_solutions_erdos_straus_equation
• elsholtz_2020_number_solutions_erdos_straus_equation / corollary_1
• elsholtz_2020_number_solutions_erdos_straus_equation / theorem_1
• elsholtz_2020_number_solutions_erdos_straus_equation / theorem_3
• elsholtz_2020_number_solutions_erdos_straus_equation / theorem_4
• erdos_1950_az_egyenlet_egesz_szamu_megoldasairol_diophantine
• erdos_1950_az_egyenlet_egesz_szamu_megoldasairol_diophantine / conjecture_p195
• mihnea_2025_further_verification_empirical_evidence_erdos_straus
• mihnea_2025_further_verification_empirical_evidence_erdos_straus / section_2
• mihnea_2025_further_verification_empirical_evidence_erdos_straus / section_3
• pomerance_2025_exceptions_erdos_straus_schinzel
• pomerance_2025_exceptions_erdos_straus_schinzel / theorem_1_1
• pomerance_2025_exceptions_erdos_straus_schinzel / theorem_1_3
• terzi_1971_conjecture_erdos_straus
• terzi_1971_conjecture_erdos_straus / table_2
• terzi_1971_conjecture_erdos_straus / verification_p215
• vaughan_1970_problem_erdos_straus_schinzel
• vaughan_1970_problem_erdos_straus_schinzel / lemma_1
• vaughan_1970_problem_erdos_straus_schinzel / lemma_2
• vaughan_1970_problem_erdos_straus_schinzel / theorem_p193