Openai 2026 short egyptian fractions

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corollary_1_2: The manuscript's claimed two-sided bound ck <= log log F(k) <= Ck for all large k, deduced from Theorem 1.1 by splitting a divisor-rich denominator and an injective padding; a claimed partial answer to Problem 148, unverified here.

corollary_1_3: The manuscript's claimed bounds exp(exp(k/600)) <= v(k) <= 1 + k^(2^(k-1)) for large k and log 2/257 <= liminf log log v(k)/k <= limsup <= log 2, from a reserved-marker greedy prefix, a direct tail construction with an explicit length coefficient and a marker-preserving padding; a claimed partial answer to Problem 293, unverified here.

theorem_1_1: The manuscript's main claim: for all b at least an absolute b_0, every a/b with 1 <= a < b is a sum of at most c_2 log log b distinct unit fractions, with the classical matching lower bound; the conjecture of Problem 304, attributed by the release to an internal model, unverified here.

OpenAI, Short Egyptian fractions, OpenAI Math Release preprint, September 25, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision adc7f1241), folder preprints/Short-Egyptian-fractions-September-25-2026; the held PDF, Short-Egyptian-fractions-September-25-2026.pdf in the release, is retained as openai_2026_short_egyptian_fractions.pdf, and the release's TeX bundle sits in the same release folder.

@misc{OAI:Short-Egyptian-fractions-September-25-2026,
  author = {{OpenAI}},
  title = {{Short Egyptian fractions}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Short-Egyptian-fractions-September-25-2026/Short-Egyptian-fractions-September-25-2026.pdf}{OAI:Short-Egyptian-fractions-September-25-2026}},
  year = {2026}
}

Attestation, as the source states it. The release's root README says that the repository holds manuscripts and proof artifacts "produced by an internal OpenAI model", that the collection "includes results at different stages of verification", that not all manuscripts have Lean formalizations, and that "Some of the unformalized results could have issues"; it describes the common procedure as three hours of thinking compute per result on average with an unreleased internal model. The manuscript's own README adds only the title, the author line "OpenAI", the date and the citation block; it carries no statement on human assistance. The manuscript names no individual author, affiliation, arXiv identifier or journal. These are the source's provenance attestations, recorded here as history, not as this corpus's review: no refereed publication, no arXiv version and no independent review of the manuscript is recorded here and nothing on this card is independently reviewed.

Formalization, as the release lists it. The release's Lean catalog (lean/formalization.yaml) names this manuscript as a source and lists three declarations as formalized main results, all in OAI/NumberTheory/EgyptianFractions/Main.lean under the namespace Problem337 (the release's namespace label; it is not the number of any problem this card bears on): main_double_log_order (the two-sided bound of Theorem 1.1), counting_double_log_order (Corollary 1.2) and prescribed_denominator_corollary (Corollary 1.3, all three clauses). The release's own Lean page for this manuscript says the formalization proves existence of an expansion for every $a/b$, the $\Theta(\log\log b)$ order of the largest minimum length, $\log\log F(k)=\Theta(k)$, a bound on every denominator of such an expansion, the occurrence of every $m\ge2$ as a denominator with the eventual length $(257/\log2+\varepsilon)\log\log m$, the padding lemma, and the bounds $e^{e^{k/600}}\le v(k)\le1+k^{2^{k-1}}$ and $\liminf\log\log v(k)/k\ge\log2/257$. It names two comparator statement files, lean/ComparatorChallenges/EgyptianFractions.lean (nine statements with sorry, matched against the Problem337 declarations by EgyptianFractions.json) and lean/ComparatorChallenges/ShortEgyptianFractions.lean (one statement, OAI.ShortEgyptian.main, existence plus the two-sided bound, matched by ShortEgyptianFractions.json against a second development under OAI/NumberTheory/ShortEgyptian/); the catalog's main-results list names only the three Problem337 declarations. All of this is read statically from the release's catalog. The corpus's verification built the Problem337 declarations main_double_log_order and egyptian_length_is_minimum (for Problem 304), counting_double_log_order and one_expansions_finite_and_bounded (for Problem 148), and prescribed_denominator_corollary and missing_denominator_semantics (for Problem 293), and checked their axioms (propext, Classical.choice and Quot.sound only). For Problem 304 that verification covers the question, answered yes: there are $c_1,c_2>0$ and $b_0$ with $c_1\log\log b\le N(b)\le c_2\log\log b$ for every $b\ge b_0$, where $N(a,b)$ is the least number of terms $1<n_1<\dots<n_k$ with sum $a/b$ and $N(b)$ is the maximum over all $1\le a<b$; so $N(b)\ll\log\log b$, in fact $N(b)\asymp\log\log b$, which also gives the order of magnitude that the problem's request to estimate $N(b)$ asks for. For Problem 148 it covers the double-exponential order of $F(k)$, the number of $k$-element sets of positive integers with reciprocal sum $1$: there are $c,C>0$ with $ck\le\log\log F(k)\le Ck$ for all large $k$, so $F(k)=\exp(\exp(\Theta(k)))$, which replaces the recorded lower bound $\exp(\exp(c'k/\log k))$, the upper half being already known (Elsholtz--Planitzer); not settled are an asymptotic formula, $F(k)$ up to constant factors, and the constant in $\log\log F(k)$ (the monograph's $c_0^{2^{k(1-\varepsilon)}}$ guess). For Problem 293 it covers the double-exponential order of $v(k)$ (the problem page's Formulation: the least $m>1$ in no $k$-term representation of $1$): for all large $k$, $e^{e^{k/600}}\le v(k)\le1+k^{2^{k-1}}$ and $\log2/257\le\liminf\log\log v(k)/k\le\limsup\log\log v(k)/k\le\log2$, with $v(k)\ge e^{e^{ck}}$ eventually for each $c<\log2/257$; so $\log\log v(k)=\Theta(k)$, which proves the $e^{e^{ck}}$ lower bound van Doorn--Tang anticipated and rules out the monograph's $2^{2^{\sqrt k}}$ alternative, while the exact slope (whether $\log\log v(k)/k\to\log2$, the monograph's $2^{2^{k(1-\varepsilon)}}$ guess) and any asymptotic for $v(k)$ are not settled. The records are kept on the claim pages of Problem 304, Problem 293 and Problem 148, not on this card; OAI.ShortEgyptian.main is not named in that record and has no build or fidelity audit recorded here.

Companions: the release groups this manuscript alone; no other manuscript of the release is listed as a companion.

Read status: claims checked for Theorem 1.1, Corollary 1.2 and Corollary 1.3, read clause by clause in the TeX source (introduction.tex, lines 16--24, 64--70 and 102--117, labels thm:main, cor:counting, cor:prescribed) on 2026-10-07, together with the statements of Proposition 2.1, Lemmas 2.2--2.3, Lemma 3.1, Lemmas 4.1--4.5, Lemma 6.1, Lemmas 7.1--7.2, Proposition 8.1 and Lemmas 8.2--8.4 that the proofs route through; the proofs were read for their structure only and no step was checked; nothing here is independently reviewed.

Contents

• Abstract and Section 1, Introduction (introduction.tex; pp. 1--4): defines $N(a,b)$ as the least $k$ with $a/b=1/n_1+\cdots+1/n_k$, $2\le n_1<\cdots<n_k$ integers, with $a/b$ not necessarily reduced and no bound on the denominators, and $N(b)=\max_{1\le a<b}N(a,b)$; states Theorem 1.1 ($c_1\log\log b\le N(b)\le c_2\log\log b$ for $b\ge b_0$, absolute constants) and places it against Erdős 1950 (Theorems 1 and 2, p. 195), Erdős--Graham 1980 (pp. 37--38), Problem 304, Vose 1985 and the length-and-denominator theorem of Tenenbaum--Yokota 1990. Defines $F(k)$, the number of increasing $k$-tuples of positive integers with reciprocal sum $1$, and states Corollary 1.2 ($ck\le\log\log F(k)\le Ck$ for $k\ge k_0$), citing Konyagin 2014, Elsholtz 2016 and Elsholtz--Planitzer 2021 for the earlier bounds. Defines $D_k$ (the integers $m\ge2$ occurring as a denominator in some $k$-term distinct expansion of $1$) and $v(k)=\min(\{2,3,\ldots\}\setminus D_k)$, and states Corollary 1.3 ($e^{e^{k/600}}\le v(k)\le1+k^{2^{k-1}}$ eventually; $\log2/257\le\liminf\log\log v(k)/k\le\limsup\log\log v(k)/k\le\log2$), citing Erdős--Graham p. 35, Problem 293 and van Doorn--Tang 2026 (Theorem 1.1, Lemma 2.1, Section 3). The outline: $S=\log b$; $O(\log S)$ greedy steps leave a remainder $A/C$ with $C$ in an exponential range in $S$; an auxiliary integer $M$ with $e^S<M=e^{O(S)}$ is built so that almost every $u/(MC)$ with $u\le X$ has an $O(\log S)$-term expansion; a multiple $gAM$ is written as a sum of two such good numerators. The descent uses the identity $u/(MQ)=1/((M/t)z)+(h/(MQz))$ with $t\mid M$, $z=\lceil Qt/u\rceil$, $h=uz-Qt$, so one unit fraction reduces the numerator to the residue of $-Qt$ modulo $u$; exceptional numerators are controlled by a uniform moment of a truncated divisor function and Hölder's inequality. The section relates the method to Erdős's 1950 factorial-divisor construction, Tenenbaum--Yokota, Croot 1999 and Martin 2000. Conventions: $\log$ natural, $O$ and $\ll$ absolute unless indicated.
• Section 2, From a dense set to every numerator (elementary.tex; pp. 5--7): Proposition 2.1 (the dense-family statement: absolute $D_M>1$, $L>0$, $S_0$ with $D_X=D_M+2$, $D_C=4D_X$ such that for $S\ge S_0$ and $e^{D_CS}\le C\le e^{2D_CS}$ there are $M$ with $e^S<M\le e^{D_MS}$ and $G\subseteq\{1,\ldots,\lfloor X\rfloor\}$, $X=e^{D_XS}$, missing at most $X/8$ integers, each $u\in G$ giving $u/(MC)$ as a sum of at most $L\log S$ unit fractions with repetitions allowed); Lemma 2.2 (removing repetitions below total $1$ without changing the count, by Takenouchi's 1921 argument); Lemma 2.3 (greedy preparation: at most $1+\lceil\log_2(\log T/\log2)\rceil$ steps reach zero or a remainder $A/C$ with $A\le a$, $T\le C<T^2$); the proof of Theorem 1.1 from Proposition 2.1 (upper bound by the two-good-numerators split with $g=\lfloor X/(AM)\rfloor$; lower bound by the Sylvester-type recurrence $d_j\le s^{2^{j-1}}$ applied to $(b-1)/b$ with $1/b$ appended, giving $\log\log b\le(1+\log2)k$).
• Section 3, A uniform divisor moment (divisors.tex; pp. 7--10): Lemma 3.1, for fixed $D,r\ge1$ and $S\ge S_0(D,r)$, with $S/(2\log S)\le\log X\le DS$, $X^{1/2}\le Y\le X$, $N\le e^{DS}$: $\sum_{1\le h\le Y}d_X(N+h)^r\le Y\exp(S^{1/4})$, where $d_X(n)$ counts divisors of $n$ up to $X$. Proof by Erdős's 1952 prime-factor splitting (a prefix $d\le\sqrt Y$ of the factorization, three ranges for the next prime), Rankin's weighting for smooth prefixes (cited to Hildebrand--Tenenbaum 1993) and an elementary $\sum_{p\le T}1/p\le e\log(1+\log T)$; the manuscript says the uniform truncated bound is proved in full in the manuscript (divisors.tex lines 18--20).
• Section 4, Divisors with small residues (residues.tex and random.tex; pp. 10--19): fixes the absolute constants $K=100$, $R=1000$, $D_0=100000$, $\eta=10^{-4}$, $D_M=(2R+2)(K+2)$, the levels $X_j=e^{D_XS}\rho^j$ with $\rho=e^{-\eta m}$, $m=\lfloor S/\log S\rfloor$, and $C_j=C$ or $1$ according as $X_j>e^S$ or not. Lemma 4.1 (the residue lemma: an integer $M$ with $e^S<M\le e^{D_MS}$ divisible by the least power of $2$ at least $S^{D_0}$, and lists $T_0,\ldots,T_R$ of $2^m$ divisors, such that at each level all but $X_je^{-c_*m}$ numerators have at least $\rho|T|/2$ entries with residue at most $X_{j+1}$, $c_*=0.001$, and every $S^{D_0}<u\le e^m$ has some entry with residue at most $u^{1-\eta}$). Construction: $T_0$ is the subset products of $m$ distinct primes in $[S^K,2S^K]$; each of $T_1,\ldots,T_R$ lists the $2^m$ products taking one prime from each of $m$ independently sampled pairs of primes in the same range; $M$ is $2^a$ times the product of all of them. Lemma 4.2 (the Erdős--Turán discrepancy inequality turns Fourier bounds into many small residues); Lemma 4.3 (a van der Corput estimate for $\sum_{n\in I}e(Z/n)$ over $I\subseteq[U,2U]$, $U^4\le|Z|\le U^B$, proved from differencing and a second-derivative test given inline, citing Graham--Kolesnik 1991); Lemma 4.4 (the mean-square Fourier bound $e^{-0.01m}$ for $T_0$ at high levels, uniform in $C$); Lemma 4.5 (the second-moment bound $\exp(-0.01w)$ for a random subset-product list, by exposing all but $2s$ sampled primes, a gcd bound and additive-character orthogonality modulo a composite $q$). The proof of Lemma 4.1 (Section 4.4) chooses one realization by Markov and union bounds: the middle levels fail with probability $o(1)$ and every terminal numerator is covered by one of the $R$ independent blocks with failure probability at most $u^{-5}$.
• Section 5, Propagating the exceptional sets (descent.tex; pp. 19--22): the proof of Proposition 2.1. Fixes $r\ge\max\{2,8K_d\}$ with $K_d=3D_X/\eta$, $\alpha=1-1/r$; expands every $u\le S^{D_0}$ in binary over the power of $2$ dividing $M$; descends terminal numerators by Lemma 4.1(3) in $O(\log S)$ steps; defines the good sets $G_j$ backwards from $G_d$ by the residue step (display (5.3)), with length at most $B_0\log S+d-j$; counts bad numerators by an indexed predecessor count ($u\mid C_jt+h$, at most $d_X(C_jt+h)$ predecessors), Lemma 3.1 and Hölder, giving the recurrence $\delta_j\le\epsilon+A\delta_{j+1}^\alpha$ with $\epsilon=e^{-c_*m}$, $A=e^{2S^{1/4}}$, unrolled from $\delta_d=0$ to $\delta_0\le d\exp(2rS^{1/4}-c_*mS^{-1/4})\to0$; $L=B_0+K_d$.
• Section 6, Counting representations of one (counting.tex; pp. 22--24): Lemma 6.1 (removing repetitions at total $1$ while keeping a denominator divisible by a fixed odd $Q>1$, with length not increasing); the proof of Corollary 1.2: Theorem 1.1 on $(Q-1)/Q$ for $Q$ the product of the first $r$ odd primes gives a distinct expansion of $1$ of length $s=O(\log r)$ with a denominator $n$ divisible by $Q$, $\tau(n)\ge2^r$; the split $1/n=1/(n+d)+1/(n+n^2/d)$ over proper divisors $d$ gives at least $2^r-2s+1$ distinct expansions of one common length; the padding $1/v=1/(v+1)+1/(v(v+1))$ on the largest denominator is injective and reaches every larger length; with $r=\lfloor e^{k/(2B)}\rfloor$ this gives $F(k)\ge2^{r-1}$. Upper bound $F(k)\le k^{2^k-1}$ from $n_i\le k^{2^{i-1}}$.
• Section 7, Preserving a prescribed denominator (prescribed.tex, lines 1--198; pp. 24--26): Lemma 7.1 (a greedy prefix that reserves $1/m$ and skips the denominator $m$: $1=1/m+\sum1/n_i+R/q$ with $j<3+\log_2\log_2T$ terms, $0\le R<2m$, and when $R>0$, $q\ge T$ and $R/q$ below $1/m$ and every $1/n_i$; $q$ is kept unreduced and each multiplier is at most the preceding denominator plus one); the qualitative deduction that Theorem 1.1 applied to $R/q$ gives a distinct expansion of $1$ containing $1/m$ with $O(\log\log m)$ terms, and a finite marked expansion for every $m\ge2$; Lemma 7.2 (padding that keeps one prescribed denominator, so $D_r\subseteq D_{r+1}$ for $r\ge3$; stated as van Doorn--Tang's Lemma 2.1 with a proof included).
• Section 8, A quantitative prescribed-denominator bound (prescribed.tex, lines 199--579; pp. 27--32): Proposition 8.1 (every $m\ge m_\varepsilon$ is an exact denominator of a distinct expansion of $1$ with at most $(257/\log2+\varepsilon)\log\log m$ terms); Lemma 8.2 (a common integer $K_m=P(m^4)^2\lfloor\log\log m\rfloor!$ with $\log K_m\le(32/\log2+o(1))\log m\log\log m$ such that every $1\le s\le m^4$ is a sum of at most $16$ rationals $e_i/t_i$ with $e_i\mid K_m$; proved through a count of exceptional primes in the style of Gallagher's larger sieve, the quantitative three-prime theorem, and a five-fold sum of products modulo $p$ resting on a bilinear exponential-sum bound recorded from Glibichuk--Konyagin 2007, with proof inline); Lemma 8.3 (the unreduced greedy denominator has a divisor in $[w/m,w]$ for every $1\le w\le q$); Lemma 8.4 (grouping: $X/(qK_m)$ is a sum of at most $B(\log X/(2\log m)+2)$ unit fractions); the proof of Proposition 8.1 ($1+j+16G\le(1+16\cdot16)/\log2+o(1)$ times $\log\log m$); the proof of Corollary 1.3 (for each fixed $c<\log2/257$, every $2\le m\le e^{e^{ck}}$ lies in $D_k$ for large $k$ by Proposition 8.1, the finitely many small markers, and Lemma 7.2 iterated; $257/\log2<600$ gives the displayed $e^{e^{k/600}}$; the upper bound from $n_i\le k^{2^{i-1}}$). A closing remark says the endpoint $c=\log2/257$ itself and any sharp slope are not asserted.
• References (pp. 32--33): 23 entries, among them Nakayama 1940, Erdős 1950, Erdős--Graham 1980, Vose 1985, Takenouchi 1921, Erdős--Turán 1948, Graham--Kolesnik 1991, Selberg 1949, van Doorn--Tang 2026, Konyagin 2014, Elsholtz 2016, Elsholtz--Planitzer 2021, Kumchev 1997, Kumchev--Tolev 2005, Tenenbaum--Yokota 1990, Erdős 1952, Hildebrand--Tenenbaum 1993, Gallagher 1971, Martin 2000, Croot 1999, Glibichuk--Konyagin 2007, and the site pages for Problems 304 and 293.

External inputs the proofs rest on, at statement level: the prime number theorem (Selberg 1949, for $P\ge S^{K-1}$ and $\log P(v)$); the Erdős--Turán discrepancy inequality (1948); the quantitative three-prime theorem with a singular series bounded below uniformly in odd $u$ (Kumchev 1997; Kumchev--Tolev 2005); Takenouchi's finiteness argument (1921); Rankin's method (Hildebrand--Tenenbaum 1993). The van der Corput estimate (Lemma 4.3), the bilinear exponential-sum bound of Glibichuk--Konyagin and van Doorn--Tang's padding lemma are cited but proved inline. The random lists of Lemma 4.1 are chosen by a probabilistic existence argument, not by computation; the manuscript flags nothing as numerical, computer-assisted or conditional. The constants $c_1,c_2,b_0$ of Theorem 1.1 and $c,C,k_0$ of Corollary 1.2 are not made explicit (the thresholds come from "sufficiently large $S$" at many places); the slope $\log2/257$ of Corollary 1.3 is explicit.

Bears on

• Problem 304: Theorem 1.1 is a claimed resolution of the exact question. The problem asks whether $N(b)\ll\log\log b$ with $N(b)=\max_{1\le a<b}N(a,b)$ over all $a$, no coprimality condition, denominators above $1$; the manuscript's $N(a,b)$ and $N(b)$ are the same quantities, and the theorem claims $N(b)\le c_2\log\log b$ for all $b\ge b_0$ with an absolute $c_2$, which would replace the page's recorded upper bound $N(b)\ll\sqrt{\log b}$ (Vose). The matching lower bound is Erdős's 1950 Theorem 2, reproved in Section 2. The claim is unverified here; the page's status rests on acceptance evidence.
• Problem 293: Corollary 1.3 is a claimed partial answer. For the page's reading of $v(k)$ (the least $m>1$ absent from every $k$-term distinct expansion of $1$, which is the manuscript's definition), it claims $\log\log v(k)=\Theta(k)$, with $v(k)\ge e^{e^{ck}}$ for every fixed $c<\log2/257$ and large $k$, hence eventually $v(k)\ge e^{e^{k/600}}$; this would replace the recorded lower bound $e^{ck^2}$ (van Doorn--Tang) and supply the doubly exponential growth their Section 3 anticipated. The corollary's upper bound $v(k)\le1+k^{2^{k-1}}$ is weaker than the page's recorded $c_0^{(2/5+o(1))2^k}$, with $c_0=1.26408\ldots$ the Vardi constant, which the manuscript acknowledges. The growth of $v(k)$ beyond its double-logarithmic order is not determined. Unverified here; the page's status rests on acceptance evidence.
• Problem 148: Corollary 1.2 is a claimed partial answer. For the page's $F(k)$ (increasing $k$-tuples with reciprocal sum $1$, the manuscript's definition), it claims $\log\log F(k)=\Theta(k)$, which would remove the $1/\log k$ from the recorded lower bound $\exp(\exp(ck/\log k))$ (Konyagin; Elsholtz). The corollary's upper bound $F(k)\le k^{2^k-1}$ is weaker than the recorded Elsholtz--Planitzer bound and is not an improvement. No asymptotic formula or constant is claimed, so "good estimates" remains open beyond the order of the double logarithm. Unverified here; the page's status rests on acceptance evidence. The manuscript cites the question to Erdős--Graham p. 32 and does not name the problem number.