Problem 206

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claims/: The 1 claim page of Problem 206, one per claimant's result; the problem's standing derives from them.

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Statement. Let $x>0$ be a real number. For any $n\geq 1$ let

$$R_n(x) = \sum_{i=1}^n\frac{1}{m_i}<x$$

be the maximal sum of $n$ distinct unit fractions which is $<x$.

Is it true that, for almost all $x$, for sufficiently large $n$, we have

$$R_{n+1}(x)=R_n(x)+\frac{1}{m},$$

where $m$ is minimal such that $m$ does not appear in $R_n(x)$ and the right-hand side is $<x$? (That is, are the best underapproximations eventually always constructed in a 'greedy' fashion?)

Formulation. "Almost all" means Lebesgue almost every $x>0$. The denominators are distinct positive integers; the maximum defining $R_n(x)$ exists (by the argument of Nathanson's Theorem 3, on the card linked below, which the paper states for $x\in(0,1]$ with repeated denominators $\ge2$), so $R_n(x)$ is a rational number that may have several representations. Kovač's paper phrases the property as the existence of one strictly increasing sequence $(m_k)$ and an $n_0$ with $\sum_{k\le n}1/m_k=R_n(x)$ for every $n\ge n_0$; this is the site's recursion, because nested best sums force the added denominator to be the least unused one that keeps the sum below $x$ and to exceed the denominators already used. The question is yes or no, and the answer is no. The companion questions for every rational $x$ and for every algebraic $x$ are variants, not the problem; they are recorded separately below.

Status. Disproved. Kovač's Theorem 1 (J. Number Theory 268 (2025), 39--48) shows that the set of $x>0$ whose best Egyptian underapproximations are eventually greedy has Lebesgue measure zero, so the assertion fails for almost every $x$ instead of holding for almost every $x$. The site's label is DISPROVED (LEAN); the suffix is a catalog label explained under Formalization. The claim page Kovač 2024 records the result, its postings and the acceptance evidence (refereed publication and the curator's credit) from which the standing above derives.

Source. erdosproblems.com/206, accessed 2026-09-17: the problem page (DISPROVED (LEAN); no last-edited date shown), its five-comment discussion thread and its empty proof-claim tab. The site cites [ErGr80, p. 31]. Cite as: T. F. Bloom, Erdős Problem #206, https://www.erdosproblems.com/206, accessed 2026-09-17.

References.

• [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. 31.
• [Cu22] Curtiss, D. R., On Kellogg's Diophantine problem. Amer. Math. Monthly 29 (1922), no. 10, 380--387.
• [Er50b] Erdős, P., Az $1/x_1+\cdots+1/x_n=a/b$ egyenlet egész számú megoldásairól (On a Diophantine equation). Mat. Lapok 1 (1950), 192--210.
• [Na23] Nathanson, M. B., Underapproximation by Egyptian fractions. J. Number Theory 242 (2023), 208--234, doi:10.1016/j.jnt.2022.07.005; arXiv:2202.00191v2 (2022).
• [Ch23b] Chu, H. V., A threshold for the best two-term underapproximation by Egyptian fractions. Indag. Math. (N.S.) 35 (2024), no. 2, 350--375, doi:10.1016/j.indag.2024.01.006; arXiv:2306.12564v2 (2024).
• [Ko24b] Kovač, V., On eventually greedy best underapproximations by Egyptian fractions. J. Number Theory 268 (2025), 39--48, doi:10.1016/j.jnt.2024.09.004; arXiv:2406.07218v3 (2024).
• [KoTa26] Kovač, V. and Tang, Q., Eventually greedy best Egyptian underapproximations of rational numbers via optimal control. arXiv:2607.28387v2 (4 August 2026), 30 pages; preprint.

Formalization. Statement only. The file ErdosProblems/206.lean of formal-conjectures at the linked revision (the main head of 2026-09-17) declares erdos_206 : answer(False) ↔ ∀ᵐ x ∂(volume.restrict (Set.Ioi (0 : ℝ))), EventuallyGreedy x under category research solved, with proof sorry, and carries a formal_proof attribute pointing to an external Lean 4 file. The community database records the formal status as Lean and no formal-proof URL. This corpus has built and audited neither file; see "Formalization and the Lean label" below.

Current assessment

The question (site formulation). The statement above; status DISPROVED (LEAN). The commentary attributes the disproof to Kovač [Ko24b] and adds: Erdős and Graham wrote that it is "not difficult" to construct irrational $x$ for which the property fails but gave no proof or reference, and no explicit such $x$ is known; the property holds for $x=1$ (Curtiss [Cu22]), for $x=1/m$, $m\ge1$ (Erdős [Er50b]), for $x=a/b$ with $a\mid b+1$ (Nathanson [Na23]) and for a larger class of rationals (Chu [Ch23b]); whether it holds for every rational $x>0$ is listed as unknown; without "eventually" it fails for some rationals, for example $R_1(11/24)=1/3$ but $R_2(11/24)=1/4+1/5$; see also Problem 282. The origin is printed p. 31 of the 1980 monograph: the claim for every rational $a/b$ is stated without proof, then "An attractive conjecture is that this also holds for any algebraic number as well. It is not difficult to construct irrationals for which the result fails. Conceivably, however, it holds for almost all reals."

Status-defining source. Kovač, Theorem 1: the set of positive reals with eventually greedy best Egyptian underapproximations has Lebesgue measure zero. Version: arXiv:2406.07218v3 (26 September 2024; the arXiv comment says v3 incorporates the referee's suggestions), and the paper appeared in J. Number Theory 268 (March 2025), 39--48 (Crossref record checked 2026-09-17); the published text was not compared with that version. Refereed publication is the acceptance evidence. The statement is checked clause by clause, and the seven-page proof is summarized for its structure on the result page: a recursive reduction to two-term underapproximations, in which Lemma 3 (for every $i\ge1000$ at least one per mille of the numbers in $(1/i,1/(i-1)]$ have non-greedy best two-term underapproximations) drives the geometric decay $|X_{s,t+2}|\le\frac{1999}{2000}|X_{s,t}|$ of Lemma 4 for the sets of numbers whose best $n$-term sums are nested for $s\le n\le t$. The proof is not rewritten in the corpus and has not been independently reviewed. Its Corollary 2 gives, non-constructively, a transcendental number without the property.

Where the property does hold (historical and current results).

• $x=1$: Curtiss, Theorem I (1922): the least positive value of $1-\sum_{k\le n-1}1/x_k$ over positive integers is $1/u_n$, where $u_1=1$ and $u_{k+1}=u_k(u_k+1)$, attained only at $x_k=u_k+1$. So $R_n(1)=\sum_{k\le n}1/(u_k+1)=1-1/u_{n+1}$ for every $n$, with denominators $2,3,7,43,\ldots$, and the best underapproximations of $1$ are greedy at every step, not only eventually. Takenouchi independently found, in a paper that had then just appeared (Proc. Phys.-Math. Soc. Japan (3) 3, 78--92), the largest unknown in solutions of $\sum1/x_i=b/a$ for $a=(m+1)b-1$, which for $b=m=1$ is Kellogg's bound, but did not prove Theorem I itself (Curtiss's author's note, p. 387).
• $x=1/m$: attributed by the site and by Kovač (p. 2: Erdős "observed that this remains to hold whenever $x=1/b$ itself is a unit fraction") to Erdős [Er50b]. What the 1950 paper states and proves is the case $x=1$, on its result page Theorems 3--5: the Sylvester sequence gives the largest proper fraction with $N(a,b)\le n$, so the best $n$-term underapproximation of $1$ is the greedy one, the statement Curtiss proved. No statement for $x=1/m$ appears on printed pp. 194--197, 203--204 and 208, and a search of the text of all nineteen pages finds none, so the $1/m$ case rests on the two attributions.
• $x=p/q\in(0,1]$ with $p\mid q+1$: Nathanson, Theorem 5 (2023): the greedy $n$-term sequence, $a_1=(q+1)/p$ and $a_{k+1}=qa_1\cdots a_k+1$, is the unique best $n$-term underapproximation for every $n$, in the convention that allows repeated denominators; since it is strictly increasing, it is also the unique best among distinct denominators.
• $x=p/q$ with $p<q$, $q$ odd and $2$ the least $\ell\ge1$ with $p\mid q+\ell$: Chu, Theorem 1.12 (2024), the "larger class" of the site's commentary. Chu's Theorem 1.3 gives the threshold for two-term greed: for $p<q$ with $\Upsilon(p,q)\le3$ (the least $j\ge1$ with $p\mid q+j$) the greedy pair is the unique best two-term underapproximation, except that $10/17$ has the two equal best pairs $1/2+1/12=1/3+1/4$, and for each $k\ge4$ some $p/q$ with $\Upsilon(p,q)=k$ has a non-greedy best pair.
• every positive rational (a preprint claim): Kovač and Tang, Theorem 1 (arXiv:2607.28387, v1 30 July 2026, v2 4 August 2026): for every rational $\lambda>0$, in both the distinct and the repeated-denominator convention and with every denominator at least $2$, there is $n_0$ such that $R_{n_0+m}(\lambda)=R_{n_0}(\lambda)+R_m(\lambda-R_{n_0}(\lambda))$ for all $m\ge1$, the greedy $m$-term tuple is the unique maximizer for the remainder, and a maximizing $n_0$-tuple can be extended by the greedy denominators. For $\lambda\le1$ this would settle the rational companion question that the site lists as unknown (for $\lambda>1$ the problem also admits the denominator $1$, which the paper's tuples exclude) and the one that Nathanson posed as Open problem (4). It is an author preprint, with no independent review found, and its Declaration of AI usage (p. 28) says that key steps of the proof were provided by OpenAI's GPT-5.6 Sol and rewritten by the authors. It is recorded as a claim with provenance, not as a status; the frontmatter status concerns the almost-all question only. The second author announced the result in the site discussion on 31 July 2026.
• algebraic $x$: open. Erdős and Graham's "attractive conjecture"; Kovač and Tang write that they cannot resolve it. A site comment of 9 August 2025 speculates that all algebraic numbers are eventually greedy; it carries no argument.
• an explicit non-example: open. Kovač's Corollary 2 is non-constructive. In the other direction, Kovač and Tang's Example 2 gives an explicit irrational, the Liouville number $\sum_{n\ge1}2^{-n!}$, whose greedy $n$-term underapproximation is the unique best one for every $n$, in both conventions, answering Nathanson's Open problem (1).

Formalization and the Lean label. The site's Lean suffix is a catalog label. The formal-conjectures file at the pinned commit is a statement with a sorry body; its formal_proof attribute names the file src/latest/ErdosProblems/Erdos206.lean of the repository plby/lean-proofs at the linked revision. That file (89,993 bytes, imports Mathlib) contains no sorry, ends with theorem erdos_206 : volume {x : ℝ | EventuallyGreedy x} = 0 and a #print axioms comment listing only propext, Classical.choice and Quot.sound; its header names Aristotle and Matteo Del Vecchio as formal authors and links the gist that the site discussion (comment of 28 April 2026) describes as an autoformalization of Kovač's proof by Aristotle. Its EventuallyGreedy has the same shape as the formal-conjectures definition (a strictly increasing sequence of positive denominators whose prefixes are best $n$-term underapproximations, quantified over all finite sets of positive naturals with sum below $x$). The comparison covers the statement only: no build, no audit of the 1,980-line proof and no kernel credit is claimed. The community database (teorth/erdosproblems) lists formal_status Lean, as of its entry's last update on 28 April 2026, and no formal-proof URL.

Search scope. None of the routes below found a refereed source changing the answer above, a published version of the rational claim, or an explicit non-example.

• The site: problem page, discussion thread and proof-claim tab (empty); formal-conjectures at the pinned commit; the community database file.
• arXiv abstract pages for 2406.07218 (v1 11 June 2024, v2 13 June 2024, v3 26 September 2024; no journal reference), 2607.28387 (v1 30 July 2026, v2 4 August 2026), 2306.12564 (v1 19 June 2023, v2 20 January 2024) and 2202.00191 (v1 1 February 2022, v2 3 February 2022).
• Crossref records for the three DOIs above and a bibliographic query for the Kovač--Tang title (no journal record).
• Semantic Scholar citation lists: three papers cite Kovač 2024 (Kovač--Tang; Li and Tang, Acta Math. Hungar. 177 (2025), whose conditional Theorem 1.6 Kovač--Tang make unconditional as their Corollary 3; and a 2025 Fibonacci Quarterly paper on two-term underapproximations), and none cites Kovač--Tang.
• arXiv API listings: "eventually greedy" (three records: the two Kovač papers and Li--Tang arXiv:2503.12277), "Egyptian underapproximation(s)" (four), "Problem 206" AND Egyptian (one).
• The primary sources [Cu22] (page images), [Na23], [Ch23b], [Ko24b], [KoTa26] and printed p. 31 of [ErGr80] (page image), read as stated.

Not searched: MathSciNet, zbMATH, Google Scholar, X. Not read: Graham's 2013 survey and the Li--Tang papers (cited through Kovač--Tang), the published versions of [Na23], [Ch23b] and [Ko24b], and the Hungarian text of [Er50b] beyond printed pp. 194--197, 203--204 and 208.

Remaining gaps. (1) Kovač's proof is compiled as statement and structure only; a full rewriting and an independent review remain outstanding. (2) The $x=1/m$ case rests on attributions: the 1950 paper's $x=1$ theorems are paged, and no $1/m$ statement was located on the pages read. (3) The rational companion is a preprint claim with a disclosed AI-assisted proof; it becomes a recorded result when a refereed version or an independent review appears. (4) Published versions of three sources are not compared with their arXiv versions. (5) No explicit non-example and no result on algebraic $x$ exist in the sources read. (6) The Lean artifacts are pointers, not local evidence.

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
• chu_2023_threshold_best_two_term_underapproximation_egyptian
• chu_2023_threshold_best_two_term_underapproximation_egyptian / theorem_1_12
• chu_2023_threshold_best_two_term_underapproximation_egyptian / theorem_1_3
• curtiss_1922_kellogg_s_diophantine_problem
• curtiss_1922_kellogg_s_diophantine_problem / theorem_i
• erdos_1950_az_egyenlet_egesz_szamu_megoldasairol_diophantine
• erdos_1950_az_egyenlet_egesz_szamu_megoldasairol_diophantine / theorem_4
• kovac_2024_eventually_greedy_best_underapproximations_egyptian_fractions
• kovac_2024_eventually_greedy_best_underapproximations_egyptian_fractions / corollary_2
• kovac_2024_eventually_greedy_best_underapproximations_egyptian_fractions / lemma_3
• kovac_2024_eventually_greedy_best_underapproximations_egyptian_fractions / lemma_4
• kovac_2024_eventually_greedy_best_underapproximations_egyptian_fractions / theorem_1
• kovac_2026_eventually_greedy_best_egyptian_underapproximations_rational
• kovac_2026_eventually_greedy_best_egyptian_underapproximations_rational / example_2
• kovac_2026_eventually_greedy_best_egyptian_underapproximations_rational / theorem_1
• li_2025_conjecture_erdos_graham_about_sylvester_s
• li_2025_conjecture_erdos_graham_about_sylvester_s / theorem_1_9
• nathanson_2023_underapproximation_egyptian_fractions
• nathanson_2023_underapproximation_egyptian_fractions / open_problem_4
• nathanson_2023_underapproximation_egyptian_fractions / theorem_5