Problem 282

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Statement. Let $A\subseteq \mathbb{N}$ be an infinite set and consider the following greedy algorithm for a rational $x\in (0,1)$: choose the minimal $n\in A$ such that $n\geq 1/x$ and repeat with $x$ replaced by $x-\frac{1}{n}$. If this terminates after finitely many steps then this produces a representation of $x$ as the sum of distinct unit fractions with denominators from $A$.

Does this process always terminate if $x$ has odd denominator and $A$ is the set of odd numbers? More generally, for which pairs $x$ and $A$ does this process terminate?

Formulation. The site's wording(the page shows no last-edited date). The step "choose the minimal $n\in A$ such that $n\ge1/x$" does not by itself exclude a denominator already used: for $x=2/3$ and $A$ the odd numbers it picks $3$ twice (discussion comment of 7 July 2026), and the site's owner replied the same day that the intended algorithm takes at each stage the largest unit fraction with an allowed denominator that has not yet been used. The formal-conjectures file models that reading (each used denominator is removed from $A$), and the 1980 monograph asks Stein's question for representations by distinct odd unit fractions (below). Pihko's published form of the question takes the greatest odd unit fraction not exceeding the remainder, which repeats a denominator only as $x_1=x_2=3$ when $x\ge2/3$ (his Remark 2.4 for $x_2=x_1$; later denominators increase strictly); under the unused-denominator rule $2/3$ gives $1/3+1/5+1/9+1/45$. So the greedy odd algorithm never repeats a denominator when $x<2/3$, and the two conventions make the same choice at every step and produce the same run for every $x<2/3$; for $x\ge2/3$ they part at the second step ($3,3,\dots$ against $3,5,\dots$), and whether they terminate on the same such inputs is not settled by the sources. The first question is Stein's odd-denominator question; the second, "for which pairs $x$ and $A$", is the general classification, of which the site's commentary names two further instances, denominators in a residue class and square denominators.

Status. Open on the site: the label is OPEN (no last-edited date shown;), and the site marks the problem as not resolvable by a finite computation. The frontmatter standing open, claim none, rests on no claim page: the site's proof-claim tab is empty, and no manuscript located claims either question. No proof, disproof or accepted resolution of the odd-denominator question, and no classification of the terminating pairs $(x,A)$, was found in the search whose scope the Current assessment records. The published sources cited state the odd-denominator question as open (Pihko 2001 and 2010, Louwsma and Martino 2023, Koizumi 2025); the results recorded are existence criteria, which say which rationals have a representation at all, and termination for prescribed finite numbers of steps, none of which decides termination in general. This is a bounded negative finding, not a certificate of openness.

Source. erdosproblems.com/282, accessed 2026-09-18: the problem page (OPEN, marked as not resolvable by a finite computation; source key [ErGr80, p. 30]; no last-edited date shown; additional thanks to Zach Hunter), its discussion thread (4 comments, 9 August 2025 to 7 July 2026) and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #282, https://www.erdosproblems.com/282, accessed 2026-09-18.

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), pp. 30--31 (the site cites p. 30). Library home: erdos_1980_old_new_problems_results_combinatorial_number_theory.
• [Gr64b] Graham, R. L., On finite sums of unit fractions. Proc. London Math. Soc. (3) 14 (1964), 193--207, DOI 10.1112/plms/s3-14.2.193. The arithmetic-progression criterion, section 4(1), p. 206; the Stewart--Breusch existence theorem recalled on p. 193. Library home: graham_1964_finite_sums_unit_fractions.
• [We65] Webb, W. A., Sums of rational numbers. Canad. J. Math. 17 (1965), 1019--1024, DOI 10.4153/cjm-1965-096-3. The Breusch--Stewart theorem recalled as background, p. 1019; Theorem 2, pp. 1020--1023. Library home: webb_1965_sums_rational_numbers.
• [MaSh21] Martin, G. and Shi, Y., An algorithm for Egyptian fraction representations with restricted denominators. Involve 18 (2025), 1--23; arXiv:2107.05076v1 (11 July 2021). Theorem 2.2 and the procedure of Section 3. Library home: martin_shi_2021_algorithm_egyptian_fraction_representations_restricted_denominators.
• [LoMa25] Louwsma, J. and Martino, J., Rational numbers with two-term odd greedy expansion. Integers 25 (2025), A46, 7 pp., DOI 10.5281/zenodo.15536292. Theorem 3, pp. 4--5; Corollary 5, pp. 5--6. Library home: louwsma_martino_2025_rational_numbers_two_term_odd_greedy_expansion.
• [Gu04] Guy, R. K., Unsolved problems in number theory. Third edition, Springer (2004), Section D11, Egyptian fractions. Library home: guy_2004_unsolved_problems_number_theory.
• [Gr64c] Graham, R. L., On finite sums of reciprocals of distinct $n$th powers. Pacific J. Math. 14 (1964), no. 1, 85--92, DOI 10.2140/pjm.1964.14.85. Theorem 4 and Corollary 1, p. 91. Library home: graham_1964_finite_sums_reciprocals_distinct_nth_powers.
• [Pi01] Pihko, J., Remarks on the "greedy odd" Egyptian fraction algorithm. Fibonacci Quart. 39 (2001), no. 3, 221--227, DOI 10.1080/00150517.2001.12428725. Open Problem 1.1, p. 221; Theorem 2.3, p. 223. Library home: pihko_2001_remarks_greedy_odd_egyptian_fraction_algorithm.
• [Pi10] Pihko, J., Remarks on the "greedy odd" Egyptian fraction algorithm II. Fibonacci Quart. 48 (2010), no. 3, 202--208, DOI 10.1080/00150517.2010.12428097 (Crossref record accessed). The open-problem sentence, p. 202; Corollary 3.6, p. 206. Library home: pihko_2010_remarks_greedy_odd_egyptian_fraction_algorithm_ii.
• [LoMa23] Louwsma, J. and Martino, J., Rational numbers with odd greedy expansion of fixed length. arXiv:2309.07280v1 (13 September 2023), 21 pp. Cited for its abstract. Library home: louwsma_martino_2023_rational_numbers_odd_greedy_expansion_fixed_length.
• [Ko25] Koizumi, J., Irrationality of the reciprocal sum of doubly exponential sequences. arXiv:2504.05933v1 (8 April 2025), 14 pp.; published as Integers 26 (2026), paper A28 (17 pp., which renumbers the results; the labels on this page are the preprint's). The odd-greedy passage, p. 7 of the preprint (journal p. 8). Library home: koizumi_2025_irrationality_reciprocal_sum_doubly_exponential_sequences.
• [Stei58] The monograph's [Stei (58)], its source for Stein's question; its bibliography (printed p. 122) gives it as S. K. Stein, personal communication, so there is no published statement by Stein to cite.

Formalization. Statement only. The file ErdosProblems/282.lean of formal-conjectures at the linked revision (main) defines greedyUnitFractionRem (A : Set ℕ) (x : ℚ) (t : ℕ) : ℚ, the remainder after $t+1$ greedy steps, each step taking sInf {n | n ∈ A ∧ 1 / x ≤ n} and continuing with A \ {n} (a used denominator is not reused), and declares erdos_282 {x : ℚ} (hx : x ∈ Set.Ioo 0 1) (hx_den : Odd x.den) : greedyUnitFractionRem {n | Odd n} x =ᶠ[atTop] 0 under category research open with proof sorry. The same file carries the variants general (an answer(sorry) for the set of terminating pairs $(x,A)$), fibonacci (termination for $A=\mathbb N$; category textbook, proof sorry), graham (denominators in a residue class $a\bmod d$ under the gcd condition; research open) and sq (square denominators; research open), and three category test lemmas that are proved (greedyUnitFractionRem_zero, greedyUnitFractionRem_one and greedyUnitFractionRem_sq_one). The community database records the problem as open and the statement as formalized (last updates of 31 August 2025 and 16 April 2026), no formal proof and no OEIS entry. The file was not built here.

Current assessment

The question (site formulation of 2026-09-18). The statement above; OPEN. The commentary recalls that Fibonacci observed in 1202 that the process terminates for every $x$ when $A=\mathbb N$, attributes the case of the odd numbers to Stein, and raises the general question in two further instances. For denominators in a residue class $a\bmod d$ it cites Graham's criterion [Gr64b], that $\frac mn$ is a sum of distinct unit fractions with such denominators exactly when $\bigl(\frac{n}{(n,(a,d))},\frac{d}{(a,d)}\bigr)=1$, and asks whether the greedy algorithm terminates whenever a representation exists. For square denominators it cites Graham's criterion [Gr64c], that $x$ is such a sum exactly when $x\in[0,\pi^2/6-1)\cup[1,\pi^2/6)$, asks the same, and records the belief of Erdős and Graham that the answer is no, and perhaps that the algorithm fails to terminate for almost every such $x$. The page points to Problem 206 (eventually greedy best underapproximations). The thread (four comments): 9 August 2025, a suggestion to ask the same for the shifted primes as the allowed set, since the rationals representable by reciprocals of shifted primes are known; 11 August 2025 (Kovač), a characterization of the non-terminating inputs and an irrationality viewpoint (below); 7 July 2026, the distinct-denominator ambiguity and the owner's reply (Formulation). The community database record (above) agrees with the label.

Origin. Printed p. 30 of the 1980 monograph, where the chapter on unit fractions opens with Fibonacci's greedy algorithm, defined there as choosing at each step the largest unit fraction $\frac1m$ not yet used that leaves a nonnegative remainder. The authors then pose Stein's question, cited to [Stei (58)]: "In representing $\frac a{2b+1}$ as a sum of distinct unit fractions of the form $\frac1{2m+1}$, does the greedy algorithm always terminate?" They note that a representation of $\frac a{2b+1}$ by distinct odd unit fractions always exists (citing [Gr (64) a], [Al-Li (63)], [Stew$_1$ (54)] and [Bre (54)]), recall Graham's general criterion from [Gr (64) a], that $\frac ab$ is a sum of distinct unit fractions with denominators of the form $pm+q$ exactly when $\bigl(\frac b{(b,(p,q))},\frac p{(p,q)}\bigr)=1$, and ask whether the greedy algorithm terminates in these cases too. The page continues with a perturbed allowed set: for $u_1=1$, $u_{n+1}=u_n(u_n+1)$ and $S=\{n>0:n\ne u_k\}$, the authors assert, without proof, that the rationals for which the greedy algorithm with denominators from $S$ fails to terminate are dense in $\mathbb R^+$ (pp. 30--31). Printed p. 31 recalls Graham's criterion [Gr (64) d] for finite sums of reciprocals of distinct squares, $\frac ab\in[0,\frac{\pi^2}6-1)\cup[1,\frac{\pi^2}6)=I$, and expects that the corresponding greedy algorithm fails to terminate for some rationals in $I$: "Perhaps this is so for almost all the rationals in $I$." The monograph states Stein's question for distinct odd unit fractions; its bibliography (printed p. 122) lists [Stei (58)] as a personal communication from S. K. Stein.

What is known (results at statement level).

• Existence criteria, not termination. Corollary 1 of Graham 1964 (p. 91) is the site's criterion for square denominators, a case of Theorem 4 for distinct $n$th powers (the paper notes that Erdős also obtained the squares criterion, unpublished). The residue-class criterion is first-hand: section 4(1) of [Gr64b] (p. 206; card graham_1964_finite_sums_unit_fractions) states that $p/q$ is a sum of distinct reciprocals of terms of an arithmetic progression exactly under the gcd condition the site quotes, and the paper recalls on p. 193 the Stewart--Breusch theorem that every rational with odd reduced denominator is a sum of distinct odd unit fractions, the case $a=2$, $b=1$ of that criterion; the card records that the paper defers the proofs of its section 4 applications to its proved Theorem 5, and that every positive remainder of the odd greedy process, having odd reduced denominator, stays representable. Webb's paper ([We65]; card webb_1965_sums_rational_numbers) recalls the Breusch--Stewart theorem as background (p. 1019) and proves in its Theorem 2 that a positive reduced rational with odd denominator is a finite sum of proper reduced fractions with distinct numerators and distinct denominators in prescribed arithmetic progressions (under coprimality conditions on the progressions and the denominator), an existence result with no control of the greedy orbit. Such criteria decide whether a representation exists; they say nothing about the path the greedy algorithm takes. Martin and Shi's algorithm ([MaSh21]; card martin_shi_2021_algorithm_egyptian_fraction_representations_restricted_denominators) lists every subset of a finite multiset of denominators whose reciprocals sum to a target (its Theorem 2.2), a tool for bounded searches and not a termination result. The library's card for Elsholtz's 2016 paper on representations of $1$ by distinct odd unit fractions (card) records counting results of the same existence kind; this page consumes no statement from it.
• The exact question in print. Pihko's Open Problem 1.1 (p. 221): "Does the greedy odd algorithm (for $b$ odd) always stop after finitely many steps?", for reduced $a/b$ with $b$ odd and the greatest odd unit fraction not exceeding the remainder at each step, cited to Guy's and Klee--Wagon's problem books. Section D11 of Guy's book ([Gu04]; card guy_2004_unsolved_problems_number_theory) poses the question, attributed to Stein, Selfridge, Graham and others, with the example $2/7=1/5+1/13+1/115+1/10465$, and records Kertesz's check for $2<m<n<120$, Wagon's $3/179$ with $19$ terms, Bailey's $3/2879$ with $21$ terms and $5/5809$, whose expansion Eppstein proved to halt with numerators running $5,6,\ldots,30,1$, and Broadhurst's $2/24631$. Remark 2.2 (p. 222): the question is equivalent to whether $1$ occurs in the sequence of numerators, and the paper compares it to the $3x+1$ problem; Example 2.1: $5/139$ stops after $19$ steps with numerators rising to $51$ before falling to $1$; Remark 2.5: for even $b$ the algorithm never stops ($1/2=1/3+1/7+1/43+\cdots$). Pihko's second paper [Pi10] (pp. 202--203) opens with "A well-known open problem is whether the greedy odd algorithm always stops after finitely many steps" and, per its abstract, constructs for each odd prime $p$ and $1<a<p$ infinitely many odd $b$ whose numerator sequence is $a,a+1,\dots,p-1,1$ (its Corollary 3.6, p. 206, claims checked on the pihko_2010_remarks_greedy_odd_egyptian_fraction_algorithm_ii card; the proof is recorded in outline only, not verified).
• Prescribed step counts. Pihko's Theorem 2.3 (p. 223; proof recorded in outline only): for every $s$ there are infinitely many reduced $a/b$ with $b$ odd for which the algorithm stops after exactly $s$ steps, built from odd Sylvester-type sequences $x_{i+1}\ge x_i^2-x_i+1$ on which the odd and the ordinary greedy algorithms agree. Louwsma and Martino (abstract): "It is an open question whether this expansion always has finitely many terms"; their paper classifies the reduced fractions with a given numerator whose odd greedy expansion has length $2$, and the rationals whose expansion has length $m$ and begins with prescribed odd denominators; their 2025 paper ([LoMa25], Theorem 3 and Corollary 5, claims checked on the card) gives the two-term rationals for each even numerator $n$ as finitely many denominator progressions, $\varphi(2n)$ of them for reduced fractions, in the convention that allows $1/1$ and repeated terms. Prescribed terminating families decide nothing about the general question.
• The irrationality viewpoint. Koizumi's preprint (p. 7) records, in its section on pseudo-greedy expansions: "It is an open problem whether the odd greedy expansion of a positive rational number with odd denominator terminates in finite steps", citing Guy's problem book, and notes (p. 9) that its Conjecture 6 on the gap sequence of the pseudo-greedy expansion "resembles the termination problem of the odd greedy expansion"; its Theorem 1 (p. 2) is a rigidity statement for sequences of positive integers whose ratios $a_n^2/a_{n+1}$ all lie within $1/3$ of a fixed $\beta\ge0$, not a result on this problem. The thread's comment of 11 August 2025 cites the paper for this connection.

Forum items (leads with provenance, not status). The comment of 11 August 2025 (Kovač) enumerates $A$ increasingly as $a(1),a(2),\dots$ and states that the greedy algorithm converges without terminating exactly for the $x$ in $G=\{\sum_{k\ge1}1/a(n_k):(n_k)\text{ strictly increasing},\ \sum_{l>k}1/a(n_l)<1/a(n_k-1)-1/a(n_k)\text{ for all }k\}$ (with $a(0)=0$, $1/0=\infty$), so that for squares the question asks whether a rational $x\in(0,1)$ is $\sum1/n_k^2$ with tails $\sum_{l>k}1/n_l^2<(2n_k-1)/((n_k-1)^2n_k^2)$; it says that a simple observation recorded under [263] gives, non-constructively, sets $A$ asymptotic to the squares for which the answer is negative, proposes Cantor-type subsets of $G$ and rational points in thick Cantor sets (as in [257]), and closes by saying that the commenter does not yet know whether this helps with any of the concrete questions. This is the commenter's argument, recorded as a lead and unverified. The comment of 9 August 2025 suggests the shifted primes as an allowed set. No proof claim exists on the tab.

Search scope. The status rests on these routes; none found a proof, a disproof or a proof claim.

• The site: problem page, discussion thread, empty proof-claim tab; the community database record; formal-conjectures 282.lean at the pinned commit.
• The primary sources at the cited passages: [ErGr80] (pp. 30--31), [Gr64c] (pp. 85--86 and 91--92), [Pi01] (pp. 221--223 and 227), [Pi10] (pp. 202--203), [Ko25] (pp. 2--3 and 7), [LoMa23] (abstract only).
• Publication records: arXiv listings of 2504.05933 (v1 only; the Integers 2026 version identified from the journal article itself) and 2309.07280 (v1 only, no journal reference); Crossref records for [Pi01], [Pi10] and [Gr64c] (found) and a bibliographic query for [LoMa23] (no record).
• arXiv API metadata search abs:"odd greedy" OR abs:"greedy odd" (one record, [LoMa23]). The API searches titles and abstracts only, so this zero is weak.

Not searched: MathSciNet, zbMATH, Google Scholar full text, X. Not held: the monograph's [Al-Li (63)], [Stew$_1$ (54)] and [Bre (54)], and Klee and Wagon's problem book.

Remaining gaps. (1) Stein's question has no publication of his own: the monograph cites a personal communication, and Guy's book attributes the question to Stein, Selfridge, Graham and others. (2) The site's unused-denominator convention and Pihko's agree on every $x<2/3$ (Remark 2.4); whether they terminate on the same inputs $x\ge2/3$ is not settled by the sources. (3) [LoMa23] is consumed at the abstract level; [Pi10] and Koizumi's published version have library cards. (4) The general question, for which pairs $(x,A)$ the process terminates, has no source beyond the monograph's dense-set remark (stated without proof) and the thread. There is no status-defining proof to compile.

Proof coverage. Nothing resolves either question. Graham's criteria and Pihko's theorem are recorded at statement level (claims checked against the sources or on the cards; the proof of Theorem 2.3 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
• graham_1964_finite_sums_reciprocals_distinct_nth_powers
• graham_1964_finite_sums_reciprocals_distinct_nth_powers / corollary_1
• graham_1964_finite_sums_reciprocals_distinct_nth_powers / theorem_3
• graham_1964_finite_sums_reciprocals_distinct_nth_powers / theorem_4
• graham_1964_finite_sums_reciprocals_distinct_nth_powers / theorem_a
• graham_1964_finite_sums_unit_fractions
• graham_1964_finite_sums_unit_fractions / remark_p206
• graham_1964_finite_sums_unit_fractions / theorem_1
• graham_1964_finite_sums_unit_fractions / theorem_4
• graham_1964_finite_sums_unit_fractions / theorem_5
• koizumi_2025_irrationality_reciprocal_sum_doubly_exponential_sequences
• koizumi_2025_irrationality_reciprocal_sum_doubly_exponential_sequences / conjecture_6
• louwsma_martino_2023_rational_numbers_odd_greedy_expansion_fixed_length
• louwsma_martino_2023_rational_numbers_odd_greedy_expansion_fixed_length / proposition_3_1
• louwsma_martino_2023_rational_numbers_odd_greedy_expansion_fixed_length / proposition_4_5
• louwsma_martino_2023_rational_numbers_odd_greedy_expansion_fixed_length / theorem_2_3
• louwsma_martino_2023_rational_numbers_odd_greedy_expansion_fixed_length / theorem_3_2
• louwsma_martino_2023_rational_numbers_odd_greedy_expansion_fixed_length / theorem_4_10
• louwsma_martino_2023_rational_numbers_odd_greedy_expansion_fixed_length / theorem_4_3
• louwsma_martino_2023_rational_numbers_odd_greedy_expansion_fixed_length / theorem_5_2
• louwsma_martino_2025_rational_numbers_two_term_odd_greedy_expansion
• martin_shi_2021_algorithm_egyptian_fraction_representations_restricted_denominators
• martin_shi_2021_algorithm_egyptian_fraction_representations_restricted_denominators / theorem_2_2
• pihko_2001_remarks_greedy_odd_egyptian_fraction_algorithm
• pihko_2001_remarks_greedy_odd_egyptian_fraction_algorithm / open_problem_1_1
• pihko_2001_remarks_greedy_odd_egyptian_fraction_algorithm / remark_2_4
• pihko_2001_remarks_greedy_odd_egyptian_fraction_algorithm / theorem_2_3
• pihko_2001_remarks_greedy_odd_egyptian_fraction_algorithm / theorem_3_5
• pihko_2001_remarks_greedy_odd_egyptian_fraction_algorithm / theorem_3_6
• pihko_2001_remarks_greedy_odd_egyptian_fraction_algorithm / theorem_3_7
• pihko_2001_remarks_greedy_odd_egyptian_fraction_algorithm / theorem_3_8
• pihko_2010_remarks_greedy_odd_egyptian_fraction_algorithm_ii
• pihko_2010_remarks_greedy_odd_egyptian_fraction_algorithm_ii / corollary_3_6
• pihko_2010_remarks_greedy_odd_egyptian_fraction_algorithm_ii / open_problem_p202
• pihko_2010_remarks_greedy_odd_egyptian_fraction_algorithm_ii / theorem_2_3
• pihko_2010_remarks_greedy_odd_egyptian_fraction_algorithm_ii / theorem_3_5
• webb_1965_sums_rational_numbers
• webb_1965_sums_rational_numbers / theorem_2