Problem 308

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claims/: The 1 claim page of Problem 308, one per claimant's result; the problem's standing derives from them.

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Statement. Let $N\geq 1$. What is the smallest integer not representable as the sum of distinct unit fractions with denominators from $\{1,\ldots,N\}$? Is it true that the set of integers representable as such has the shape $\{1,\ldots,m\}$ for some $m$?

Statement (corrected). Let $N$ be sufficiently large and $m_N=\lfloor \sum_{n\leq N}\frac{1}{n}\rfloor$. Is the smallest integer not representable as the sum of distinct unit fractions with denominators from $\{1,\ldots,N\}$ equal to $m_N$ or $m_N+1$? Is it true that the set of integers representable as such has the shape $\{1,\ldots,m\}$ for some $m$?

Notes. The site's wording, with "Let $N\geq 1$", asks for the exact value of the smallest non-representable integer $f(N)$ for every $N$, and whether the representable integers form an initial segment for every $N$. That is what Erdős and Graham printed: [ErGr80], p. 39, for $S_n$ the set of integers of the form $\sum_{k=1}^r\frac1{x_k}$ with $1\le x_1<\cdots<x_r\le n$ and $r$ variable, asks "What is the smallest integer not in $S_n$? Is it true that $m\notin S_n$ implies $m+1\notin S_n$?", with no range on $n$. The site reads the problem through Croot's theorem: its label is PROVED, its commentary says the problem "was essentially solved by Croot [Cr99]" and concludes that, with $m_N=\lfloor\sum_{n\le N}1/n\rfloor$, the representable integers are "for all $N$ sufficiently large, either $\{1,\ldots,m_N-1\}$ or $\{1,\ldots,m_N\}$"; the statement file it links as its formalization (formal-conjectures 308.lean, commit 9d259649abe0b02d7a25f7589b872db679b35e21) states the problem as parts.i, $f(N)\in\{m_N,m_N+1\}$ for all large $N$, and parts.ii, the initial-segment property for all large $N$, marks both solved, and keeps the property for every $N\ge1$ as a separate open variant all_N; the site's thread has no comments. The change replaces "Let $N\geq 1$" by "Let $N$ be sufficiently large and $m_N=\lfloor\sum_{n\le N}\frac1n\rfloor$", in the site's own notation, and replaces "What is the smallest integer not representable ...?" by the question whether it equals $m_N$ or $m_N+1$; the second question is unchanged. The answer under the site's reading is yes to both, by Croot's Main Theorem (Mathematika 46 (1999), 359--372; typescript p. 1), which places $n(N)=f(N)-1$ between $\lfloor H_N-\frac92(1+o(1))(\log\log N)^2/\log N\rfloor$ and $\lfloor H_N-\frac12(1+o(1))(\log\log N)^2/\log N\rfloor$ for all large $N$, so that $\{1,\ldots,m_N-1\}\subseteq N(N)\subseteq\{1,\ldots,m_N\}$ and $f(N)\in\{m_N,m_N+1\}$; the site's two displays attach these floors to $f(N)$ where the theorem bounds $n(N)$, an off-by-one in the displays that does not affect the site's consequence sentence. Under the readings the page does not adopt: the exact value of $f(N)$ is decided by the fractional part $\delta_N$ of $H_N$ outside the window $(\frac12+o(1))(\log\log N)^2/\log N<\delta_N<(\frac92+o(1))(\log\log N)^2/\log N$ (the upper threshold lowered to $\frac{\pi^2}3$ by the deduction from Yokota's 2002 Corollary 1 on its result page) and open inside it, where Croot's conjecture (p. 2) that the upper floor is the truth would close it; and whether $N(N)$ is an initial segment for every $N\ge1$ is open, asserted by no source, encoded by OEIS A101877 (the least largest denominator for each integer, nondecreasing in its eight printed terms) and marked research open by the formal-conjectures file. Both stay in Formulation with no claim page. Results about the site's wording, credited and never counted: none; Boris Alexeev's Erdos308.lean (plby/lean-proofs, commit 8822f7ddef30fadbd92e1c6ab4ed897af356af5e, 15 September 2026) states the two-case statement for large $N$, which is the corrected Statement, and is unbuilt here. The page's standing judges the corrected Statement.

Formulation. Erdős and Graham's exact questions ([ErGr80], p. 39) ask, with no range on $n$, for the smallest integer not in $S_n$ and whether $m\notin S_n$ implies $m+1\notin S_n$; read exactly, both are open, and this page records them as the problem's two variants below. Write $N(N)$ for the set of positive integers of the form $1/n_1+\cdots+1/n_k$ with $1\le n_1<\cdots<n_k\le N$ and $k$ variable, $f(N)$ for the smallest positive integer not in $N(N)$, $H_N=\sum_{n\le N}1/n$ and $m_N=\lfloor H_N\rfloor$. Every element of $N(N)$ is at most $H_N$, so $N(N)\subseteq\{1,\ldots,m_N\}$ and $f(N)\le m_N+1$. The corrected Statement asks, for all large $N$, whether $f(N)\in\{m_N,m_N+1\}$ and whether $N(N)$ is an initial segment $\{1,\ldots,m\}$, in which case $f(N)=m+1$; Croot (p. 1) likewise reads the questions of Erdős and Graham as asking about large $N$. The first variant asks for the exact value of $f(N)$. It is decided by the fractional part $\delta_N$ of $H_N$ outside the window $(\frac12+o(1))(\log\log N)^2/\log N<\delta_N<(\frac92+o(1))(\log\log N)^2/\log N$, whose upper threshold the deduction from Yokota 2002 lowers from $\frac92$ to $\frac{\pi^2}3$, and it is open inside the window, where Croot's conjecture (p. 2) that the upper floor is the truth would close it. The second variant asks whether $N(N)$ is an initial segment for every $N\ge1$, not only for large $N$. No source asserts it; the formal-conjectures statement file marks it research open as erdos_308.variants.all_N, and OEIS A101877 (the least largest denominator for each $n$) encodes it, $N(N)$ being an initial segment for every $N$ exactly when that sequence is nondecreasing; its eight printed terms are, and the sequence is a data lead, not a result. Neither variant has a claim page. The site's commentary attaches Croot's two floors to $f(N)$; Croot's theorem bounds $n(N)=f(N)-1$, the largest integer with every integer up to it representable, as recorded below.

Status. Proved, in the site's label, which the site itself calls an essential solution; the standing on this page agrees, for the corrected Statement. The one claim page, Croot's Main Theorem (Mathematika 46 (1999), no. 2, 359--372; refereed; credited by the site), is an accepted full claim: it answers both questions of the corrected Statement. The theorem, in the author's typescript, gives for all large $N$

$$\Bigl\lfloor H_N-\tfrac92(1+o(1))\tfrac{(\log\log N)^2}{\log N}\Bigr\rfloor \le n(N)\le \Bigl\lfloor H_N-\tfrac12(1+o(1))\tfrac{(\log\log N)^2}{\log N}\Bigr\rfloor.$$

Hence, for all large $N$, $N(N)$ is $\{1,\ldots,m_N-1\}$ or $\{1,\ldots,m_N\}$ (the second question is answered yes), and $f(N)\in\{m_N,m_N+1\}$ (the first question is answered yes), with the case decided by the fractional part of $H_N$ except when it lies between $(\frac12+o(1))$ and $(\frac92+o(1))$ times $(\log\log N)^2/\log N$. The value inside that window, which Croot's conjecture (p. 2) would close, and the initial-segment property for every $N\ge1$ are the two variants the Formulation records. The site's discussion and proof-claim pages carry no further claim.

Source. erdosproblems.com/308, accessed 2026-09-18: the problem page (PROVED, with the site's note that it is solved in the affirmative; source key [ErGr80]; no last-edited stamp), its empty discussion thread and its empty proof-claim tab. The site cites [Cr99] in its commentary and thanks one contributor by name. Cite as: T. F. Bloom, Erdős Problem #308, https://www.erdosproblems.com/308, accessed 2026-09-18.

References.

• [Cr99] Croot, III, Ernest S., On some questions of Erdős and Graham about Egyptian fractions. Mathematika 46 (1999), no. 2, 359--372, DOI 10.1112/S0025579300007828 (Crossref record); locators are pages of the author's 14-page typescript. Library home: crootiii_1999_questions_erdos_graham_about_egyptian_fractions; result pages main_theorem, corollary, conjecture_p2.
• [Yo97] Yokota, Hisashi, On number of integers representable as a sum of unit fractions. II. J. Number Theory 67 (1997), no. 2, 162--169, DOI 10.1006/jnth.1997.2187, with a Corrigendum, J. Number Theory 72 (1998), 150. Croot cites the Corrigendum ([6] in the paper) for the range $\{n:1\le n\le\log x-5\log\log x\}\subseteq N(x)$ (typescript p. 1), and the paper with its Corrigendum, [5] and [6], in the proof (p. 12). Of the 1997 paper (printed pp. 162--169), Theorem 1 and the opening of its proof are recorded; the Corrigendum is not held. Library home: yokota_1997_number_integers_representable_sum_unit_fractions_ii; result page theorem_1.
• [Yo02] Yokota, Hisashi, On the number of integers representable as sums of unit fractions. III. J. Number Theory 96 (2002), no. 2, 351--372, DOI 10.1006/jnth.2002.2797 (Crossref record); printed pp. 351--372, the statements recorded. Library home: yokota_2002_number_integers_representable_sums_unit_fractions_iii; result pages theorem_1, corollary_1.
• [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), printed pp. 39--40. Library home: erdos_1980_old_new_problems_results_combinatorial_number_theory.

Formalization. A statement file and an external Lean proof, neither built in this corpus. Before the statement file was added, the site's page showed "Formalised statement? No (create one)", there was no ErdosProblems/308.lean in formal-conjectures (main), and the community database (teorth/erdosproblems, data/problems.yaml) recorded status proved since 31 August 2025, formal status unformalized, statement not formalized, OEIS "possible". The statement file was added on 20 September 2026: at the linked commit, FormalConjectures/ErdosProblems/308.lean splits the problem into erdos_308.parts.i (for all large $N$, $f(N)=m_N$ or $m_N+1$), erdos_308.parts.ii (the answer is yes: for all large $N$ the representable integers form an initial segment) and erdos_308.variants.shape (the two-set alternative), each by sorry with a formal_proof annotation pointing at src/latest/ErdosProblems/Erdos308.lean in Boris Alexeev's lean-proofs collection at the linked commit of 15 September 2026; it states Croot's two floors as a further sorry variant (research solved, no pointer) and marks erdos_308.variants.all_N, the initial-segment property for every $N\ge1$, as research open. As of 2026-10-07 the site's page shows the statement as formalized and links the statement file, and the community database records the problem as formalized since 20 September 2026. The Alexeev file declares itself in its header a Lean formalization of a solution to Problem 308 with Croot and Yokota as informal authors and Codex, GPT-5.6 Sol (OpenAI Codex) as formal authors; its theorem erdos_308 is the two-case statement for all large $N$, and it is linked as a formalization from Croot's claim page. Its top file is recorded; its imports are not. This corpus has built neither file, so no formalized evidence is listed.

Current assessment

The question (site formulation). The statement above; status PROVED; source key [ErGr80]. The commentary credits Croot [Cr99] with an essential solution and displays, for $f(N)$ the smallest non-representable integer, the two floors

$$\Bigl\lfloor\sum_{n\le N}\tfrac1n-\tfrac92(1+o(1))\tfrac{(\log\log N)^2}{\log N}\Bigr\rfloor\le f(N) \quad\text{and}\quad f(N)\le\Bigl\lfloor\sum_{n\le N}\tfrac1n-\tfrac12(1+o(1))\tfrac{(\log\log N)^2}{\log N}\Bigr\rfloor$$

and concludes that, with $m_N=\lfloor\sum_{n\le N}\frac1n\rfloor$, the representable integers form the set $\{1,\ldots,m_N-1\}$ or $\{1,\ldots,m_N\}$ for every large $N$. The thread and the proof-claim tab are empty. The external-database panel and the community database record list OEIS A101877 beside "possible". A101877 is a data lead for the all-$N$ variant: its entry defines $a(n)$ as the least possible largest denominator of a set of distinct unit fractions summing to $n$ and prints eight terms, $1,6,24,65,184,469,1243,3231$, with bounds for $a(9)$ to $a(11)$ (the terms as printed, not verified). Since $N(N)=\{n:a(n)\le N\}$, the set $N(N)$ is an initial segment for every $N\ge1$ exactly when $a$ is nondecreasing, which the printed terms satisfy; the sequence encodes the all-$N$ variant of the Formulation, and no proof that it is nondecreasing is recorded.

Origin. Printed p. 39 of the 1980 monograph: "Consider the set $S_n$ of all integers which can be written in the form $\sum_{k=1}^r\frac{1}{x_k}$ with $1\le x_1<\cdots<x_r\le n$, $r$ variable. What is the smallest integer not in $S_n$? Is it true that $m\notin S_n$ implies $m+1\notin S_n$?" The page continues with a construction of $n$-element sets whose reciprocal subsums represent more integers than $\{1,\ldots,n\}$ does, and p. 40 opens with the counting question of Problem 309. Croot's reference [1] cites the monograph's pp. 39--40 and 103; printed p. 103 concerns other problems.

Status support. The status-defining source is Croot's Main Theorem (typescript p. 1; claims checked). It defines $n(x)$ as the largest integer such that every integer $1\le n\le n(x)$ is a sum of distinct unit fractions with denominators at most $x$, and proves

$$\Biggl\lfloor\sum_{n\le x}\frac1n-\frac92\,\frac{(\log\log x)^2(1+o(1))}{\log x}\Biggr\rfloor \le n(x)\le \Biggl\lfloor\sum_{n\le x}\frac1n-\frac12\,\frac{(\log\log x)^2(1+o(1))}{\log x}\Biggr\rfloor.$$

Since $\{1,\ldots,n(x)\}\subseteq N(x)$ and $n(x)+1\notin N(x)$, the smallest non-representable integer is $f(x)=n(x)+1$. The site's two displays are Croot's two floors attached to $f(N)$ instead of $n(N)$; for $f(N)$ each floor is one more, and in the case where the fractional part of $H_N$ is below $(\frac12+o(1))(\log\log N)^2/\log N$ the site's upper display reads $f(N)\le m_N-1$ while the theorem gives $f(N)=m_N$. The site's consequence sentence is unaffected: Croot writes on p. 2 that $N(x)\subseteq\{1,\ldots,m\}$ trivially, that the Main Theorem gives $\{1,\ldots,m-1\}\subseteq N(x)$ for large $x$, that $m\in N(x)$ when $\delta>(\frac92+o(1))(\log\log x)^2/\log x$ and $m\notin N(x)$ when $\delta<(\frac12+o(1))(\log\log x)^2/\log x$, where $H_x=m+\delta$ (conjecture_p2). So for all large $N$ the representable integers form an initial segment (the corrected Statement's second question, yes) and the smallest missing integer is $m_N$ or $m_N+1$ (its first question, yes); which of the two is undetermined only when $\delta_N$ lies in the window between the two thresholds (the exact-value variant). Croot conjectures that the upper bound is the truth, with the threshold $D(x)=(\frac12+o(1))(\log\log x)^2/\log x$. Acceptance evidence: publication in Mathematika, a refereed journal (the Crossref record); the site's acceptance. Proof coverage: the proof (Sections 2--7 of the typescript) is sketched in outline only on the result page; the lower bound takes the integers below a fixed bound from "the main result in [5] (and [6])", Yokota's 1997 paper with its 1998 Corrigendum [Yo97] (typescript p. 12); the 1997 paper is recorded at statement depth (below), and the Corrigendum is not held. The journal text of Croot's paper was not compared with the typescript.

Earlier and adjacent results. Yokota's Theorem 1 [Yo97] (printed p. 162; claims checked) states that for all $n>n_0$, $(1-\frac{5\log_2n}{\log n})\le\frac{|N(n)|}{\log n}<(1+\frac1{\log n})$, $\log_2$ the iterated logarithm, and its proof opens (p. 167) by stating what it shows: "every positive integer $a$ is in $N(n)$ if $a\le\log n(1-\varepsilon(n))$ with $\varepsilon(n)\le5\log_2n/\log n$ for $n$ sufficiently large." This is the range $\{n:1\le n\le\log x-5\log\log x\}\subseteq N(x)$ that Croot's introduction reports, crediting the Corrigendum ([6] in Croot's paper); it would put the smallest missing integer $f(N)$ above $\log N-5\log\log N$ for large $N$, but the printed last step (p. 168) reaches only the integers up to $\log N-(3+o(1))\log N/\log\log N$ (see the library's result page). The proof (pp. 167--169) is recorded in outline only, and the 1998 Corrigendum is not held, so what it corrects is not recorded. The paper's introduction (p. 162) records Erdős's question for "the smallest integer not in $N(n)$", the exact-value variant, citing Guy's Unsolved Problems in Number Theory. Croot's Corollary is the inverse form: every positive integer $n$ is a sum of distinct unit fractions with denominators at most $e^{n-\gamma}\{1+(\frac92+o(1))\log^2n/n\}$. Yokota's Corollary 1 [Yo02] (printed p. 353; claims checked; the proof recorded in outline only) sharpens this inverse form: with $F(a)$ the least $n$ such that $a\in N(n)$, $F(a)\le\exp[a-\gamma+(\frac{\pi^2}3+o(1))(\log a)^2/a]$ for all large $a$. A deduction written on that result page, not stated in the paper, turns this into $\{1,\ldots,\lfloor H_N-(\frac{\pi^2}3+o(1))(\log\log N)^2/\log N\rfloor\}\subseteq N(N)$ for large $N$, so the lower floor of Croot's window holds with $\frac{\pi^2}3$ in place of $\frac92$; the upper floor and the two-case residue are unchanged. The counting question for $N(N)$ is Problem 309; OEIS A217693, linked there, lists the count of representable integers for $n\le87$ or so (a community record, taken as printed). Bettin, Grenié, Molteni and Sanna's count of all Egyptian fractions with denominators at most $N$ (arXiv:1906.11986) cites Croot's paper and concerns Problem 320, not this question.

Search scope. The problem, discussion and proof-claim pages; the community database record; the formal-conjectures directory listing and full tree of main before 20 September 2026 (no file); the Crossref bibliographic record of Croot's Mathematika paper and of Yokota's 1997 and 2002 papers; the Semantic Scholar citation list of Croot's paper (fourteen records: Yokota 2002, Croot 2001, Martin 2000, Bettin et al. 2019 and 2025, survey chapters, a 2006 paper on residues modulo a prime; none closing the residue, and Yokota 2002 narrows it as recorded above); arXiv API searches for abstracts naming unit fractions and "representable" (twelve records, none on this question) and Egyptian fractions with integers, denominators and "distinct" (four, none on it), and the sixty most recent abstracts mentioning unit or Egyptian fractions (none on this problem); OEIS A217693; the zbMATH Open record list for Yokota's unit-fraction papers (three records, 1990, 1997, 2002); the primary sources [Cr99] and [ErGr80] pp. 39--40. Not searched: MathSciNet, Google Scholar, X. Nothing found closes the exact-value variant or disputes Croot's theorem. The scope also covered the formal-conjectures statement file and the Alexeev Lean file at the commits the Formalization links pin, the community database record (formalized since 20 September 2026; OEIS A101877), the site's formalization and external-database panels, and the OEIS entry A101877.

Remaining gaps. (1) The site's displays are off by one against the theorem's quantity; recorded on this page, not corrected on the site. (2) Croot's proof is compiled as a statement with a structural sketch; its small-integer input, Croot's "[5] (and [6])" (Yokota 1997, Theorem 1, with its 1998 Corrigendum), is recorded at statement depth for the 1997 paper, whose printed last step does not reach the range Croot cites, and the Corrigendum is not held. The two variants the Formulation records, the exact value of $f(N)$ inside Croot's window and the initial-segment property for every $N\ge1$, are open.

Progress and known results

• Trivial: $N(N)\subseteq\{1,\ldots,m_N\}$, so $f(N)\le m_N+1$.
• Yokota's Theorem 1 (J. Number Theory 1997, printed p. 162, with the initial-segment form its proof states on p. 167): for large $N$ every positive integer up to $\log N-5\log\log N$ is representable, which would give $f(N)>\log N-5\log\log N$, but its printed last step (p. 168) reaches only $\log N-(3+o(1))\log N/\log\log N$ (the proof recorded in outline only; the 1998 Corrigendum not held).
• Croot's Main Theorem (Mathematika 1999): the two floors for $n(N)=f(N)-1$; hence, for large $N$, $N(N)=\{1,\ldots,m_N-1\}$ or $\{1,\ldots,m_N\}$ and $f(N)\in\{m_N,m_N+1\}$, decided by the fractional part of $H_N$ outside the window between $(\frac12+o(1))(\log\log N)^2/\log N$ and $(\frac92+o(1))(\log\log N)^2/\log N$. Croot's conjecture (p. 2) names the threshold $(\frac12+o(1))(\log\log N)^2/\log N$.
• Yokota's Corollary 1 (J. Number Theory 2002, printed p. 353): $F(a)\le\exp[a-\gamma+(\frac{\pi^2}3+o(1))(\log a)^2/a]$ for the least $n$ with $a\in N(n)$, sharpening Croot's Corollary; by the deduction on its result page, $n(N)\ge\lfloor H_N-(\frac{\pi^2}3+o(1))(\log\log N)^2/\log N\rfloor$ for large $N$, narrowing the window's upper threshold (the proof recorded in outline only).
• Related: Problem 309 (the count of representable integers), Problem 320 (the count of all distinct subsums of $\{1/n:n\le N\}$).

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
• crootiii_1999_questions_erdos_graham_about_egyptian_fractions
• crootiii_1999_questions_erdos_graham_about_egyptian_fractions / conjecture_p2
• crootiii_1999_questions_erdos_graham_about_egyptian_fractions / corollary
• crootiii_1999_questions_erdos_graham_about_egyptian_fractions / main_theorem
• yokota_1997_number_integers_representable_sum_unit_fractions_ii
• yokota_1997_number_integers_representable_sum_unit_fractions_ii / theorem_1
• yokota_2002_number_integers_representable_sums_unit_fractions_iii
• yokota_2002_number_integers_representable_sums_unit_fractions_iii / corollary_1
• yokota_2002_number_integers_representable_sums_unit_fractions_iii / theorem_1