Yokota 1997 number integers representable sum unit fractions ii

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lemma_4: Lemma 4 of Yokota's 1997 paper, quoted from Theorem 1 of the author's 1990 paper: if a lies between the sums of 1/d over the divisors d of a fixed product up to p_k and up to p_(k+1), then p_k is at most exp((a − 1)/(1 − 1/log σ_t − 3/log² σ_t)); the proof of Theorem 1 uses it to bound the largest denominator.

theorem_1: Yokota's 1997 theorem: for large n, the number of integers that are sums of reciprocals of distinct integers at most n is at least log n − 5 log log n and less than log n + 1, so it is asymptotic to log n; the proof opens by stating that every positive integer up to log n − 5 log log n is such a sum, though its printed last step does not reach that range.

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H. Yokota, On Number of Integers Representable as a Sum of Unit Fractions, II, Journal of Number Theory 67 (1997), no. 2, 162--169, Article No. NT972187 (both printed on p. 162, with the copyright line "1997 by Academic Press"); DOI 10.1006/jnth.1997.2187 (the publisher's record; the DOI is not printed on the page); the author at the Department of Mathematics, Hiroshima Institute of Technology; communicated by Alan C. Woods; received July 29, 1996 (p. 162). Cited as [Yo97] on the problem pages. It is the second paper of a series: its reference 5 is the author's Part I, On number of integers representable as sums of unit fractions, Canad. Math. Bull. 33 (1990), 235--241 (not held), and its Part III is the 2002 paper filed as yokota_2002_number_integers_representable_sums_unit_fractions_iii. A Corrigendum, J. Number Theory 72 (1998), 150 (Croot's reference [6] and the 2002 paper's reference 11), is not held, so what it corrects is unknown here; every statement on this card and its result page is the 1997 printing. Its reference 1 is the 1980 Erdős--Graham monograph, cited for pp. 30--44, filed as erdos_1980_old_new_problems_results_combinatorial_number_theory; its reference 2 is Guy's Unsolved Problems in Number Theory (2nd ed., 1991), cited for Erdős's questions on the largest integer in $N(n)$ and the smallest integer not in it. Croot's Mathematika 46 (1999) paper, filed as crootiii_1999_questions_erdos_graham_about_egyptian_fractions, credits the range $\{n:1\le n\le\log x-5\log\log x\}\subseteq N(x)$ to the Corrigendum, its reference [6] ("Recently, in [6], Yokota showed that", typescript p. 1), and uses "the main result in [5] (and [6])", this paper with its Corrigendum, in the proof of its Main Theorem for the integers below a fixed bound (typescript p. 12).

The copy read for this card is the publisher's production PDF of the journal article: 8 pages, printed pp. 162--169 = PDF pp. 1--8 (printed p. $n$ is PDF p. $n-161$), distilled from the publisher's composition system on 25 November 1997 (Acrobat Distiller 3.0 per the file's metadata, whose title field misprints "One Number of Integers"; each page carries a composition foot line), with a text layer that reads the prose cleanly and garbles the mathematics (inequality signs, minus signs, Greek letters, the product and sum signs and the fraction layout come out as substitute characters, so every display was read on the page image). Provenance: the copy was obtained on 2026-09-22 as a free copy from the publisher's open archive, the DOI https://doi.org/10.1006/jnth.1997.2187 resolving to the article's PDF on the publisher's site (PII S0022314X97921879) under the publisher's user license; 246,224 bytes. The PDF prints "Copyright © 1997 by Academic Press" and, on the next line, "All rights of reproduction in any form reserved." on its first page (printed p. 162), every other right reserved.

Read status: claims checked for the abstract, the definition of $N(n)$, the trivial upper bound, the recalled question of Erdős and Graham and the author's 1990 bound, Erdős's questions and Theorem 1 (p. 162), the notation of § 2 and the statements of Lemmas 1--3 (p. 163), the statements of Lemmas 4 and 5 (p. 164), the opening sentence of § 3 with the choice of $t$, $d_0$, $k$ and $d^*$ (p. 167), the closing steps of the proof (pp. 168--169) and the reference list (p. 169), each read clause by clause on the page images of PDF pp. 1--3 and 6--8 on 2026-09-22. The proofs of Lemma 3 (pp. 163--164) and Lemma 5 (pp. 164--167) were read in the text layer for structure only, with the pages 163, 164 and 167 also on the page image; PDF pp. 4--5 (printed pp. 165--166) were read in the text layer only. No estimate was checked except the last step of the proof (p. 168, on the page image on 2026-10-07; see Contents), and nothing here is independently reviewed.

Contents

• Abstract and § 1, Introduction (p. 162, page image). "Let $N(n)$ be the set of integers that can be written as a sum of distinct reciprocal of integers $\le n$. Then $|N(n)|\sim\log n$, which gives the correct order of $|N(n)|$." The introduction writes $N(n)$ as the set of all integers $\sum_{i=1}^n\varepsilon_i/i$ with $\varepsilon_i\in\{0,1\}$, notes the trivial upper bound $|N(n)|\le\log n+1$, recalls that Erdős and Graham [1] "asked whether $|N(n)|=O(\log n)$", that the author [5] settled it with $|N(n)|\ge(\frac12-\varepsilon(n))\log n$, $\varepsilon(n)\to0$, and that Erdős [2] asked for the size of the largest integer in $N(n)$, of the smallest integer not in $N(n)$, and for the number of integers below $\sum_1^n1/i$ not of the form $\sum_i^n\varepsilon_i/i$: "these questions can be answered if we can give the correct order of $|N(n)|$." Theorem 1 is quoted on its result page. Here $\log_j$ is the $j$-fold iterated logarithm, so $\log_2n=\log\log n$ (the introduction does not define it; § 2 and Lemma 1 use $\log_2$ and $\log_3$ in this sense).
• § 2, Lemmata (pp. 163--167). $p$, with or without subscript, is a prime, $p_j$ the $j$th prime, and $S=\{\sigma_j\}$ the increasing sequence of all positive integers $p^{2^i}$, $i\ge0$ (p. 163). Lemma 1 (p. 163): for large $k$ and $\prod_{j<k}p_j<N<\prod_{j\le k}p_j$, $p_k\le\log N(1+2/\log_2N)$ and $k\le\frac{\log N}{\log_2N}(1+\frac{\log_3N}{\log_2N})$, "a simple consequence of prime number theory". Lemma 2 (p. 163): for $t\ge t_0$ and $(1-2/\sqrt{\sigma_t})\prod_1^t\sigma_i<r<2\prod_1^t\sigma_i$, $r$ is a sum of $m\le\pi(\sigma_t)+2\pi(\sqrt{\sigma_t})\log\sigma_t$ distinct divisors $d_i$ of $\prod_1^t\sigma_i$ with $d_i\ge\prod_1^t\sigma_j/2\sigma_t^2\log\sigma_t$, quoted as Lemma 2.7 of the author's 1988 paper [4]. Lemma 3 (pp. 163--164): for $t\ge t_0$ and a prime $p$ with $\prod_1^{t-1}\sigma_i<p<\prod_1^t\sigma_i$, there are $m\le\pi(\sigma_t)+2\pi(\sqrt{\sigma_t})\log\sigma_t$ distinct divisors $d_i\le2\sigma_t^2\log\sigma_t$ of $\prod_1^t\sigma_i$ such that $\{\prod_1^t\sigma_i\sum_1^m\varepsilon_i/d_i:\varepsilon_i\in\{0,1\}\}$ is a complete residue system modulo $p$; proved from Lemma 2 by writing $a/p=(ps+r)/p\prod_1^t\sigma_i$. Lemma 4 (p. 164): for $a$ with $\sum_{d\le p_k}1/d<a<\sum_{d\le p_{k+1}}1/d$ over $d\mid\prod_1^t\sigma_i\prod_u^kp_j$, $p_k\le\exp(\frac{a-1}{1-1/\log s_t-3/\log^2s_t})$ (the printed $s_t$ is the $\sigma_t$ of the rest of the paper), quoted as Theorem 1 of the author's 1990 paper [5]. Lemma 5 (p. 164): for $t\ge t_0$, $p_k<\prod_1^{t-1}\sigma_i$ and $r$ between $(1+\frac{\log\sigma_t\log_2p_k}{\sigma_t}-\frac1{\sqrt{\sigma_t}})\prod_1^t\sigma_i\prod_u^kp_j$ and $(2+\frac{\log\sigma_t\log_2p_k}{\sigma_t}-\frac1{\sqrt{\sigma_t}})\prod_1^t\sigma_i\prod_u^kp_j$, $r$ is a sum of $m\le\pi(\sigma_t)+2\pi(\sqrt{\sigma_t})\log\sigma_t+2p_k$ distinct divisors $d_i$ of $\prod_1^t\sigma_i\prod_u^kp_j$ with $d_i\ge\prod_1^t\sigma_j\prod_u^kp_j/2p_k\sigma_t^3\log\sigma_t$. Its proof (pp. 165--167, text layer) peels the primes $p_k,p_{k-1},\ldots,p_u$ off one at a time, at each step using the complete residue system of Lemma 3 (scaled to divisors $d_j^*$ of the product) to reduce $r$ modulo $p_j$, applies Lemma 2 to the remainder $r_{k-u+1}$ between $(1-1/\sqrt{\sigma_t})\prod_1^t\sigma_i$ and $2\prod_1^t\sigma_i$, checks that the divisors produced are distinct and large enough, and bounds the count $m$ by Mertens's first theorem [3].
• § 3, Proof of Theorem 1 (pp. 167--169; pp. 167--169 on the page images). "We show that 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." For a large integer $a$: choose $t$ with $a\le\sigma_t<2a$, $d_0$ the smallest $p^{2^j}$ ($j\ge1$) above $\sigma_t$, $k$ with $\sum_{d\le p_k}1/d<a-1/d_0<\sum_{d<p_{k+1}}1/d$ over the divisors $d$ of $\prod_1^t\sigma_i\prod_u^kp_j$ ($p_u$ the smallest prime $\ge\sigma_t$), and $d^*$ the largest such divisor with $\sum_{d\le d^*}1/d<a-1/d_0$. With $d^-(d^*)$ the next divisor below $d^*$, the deficit $a-1/d_0-\sum_{d\le d^-(d^*)}1/d$ lies between $1/d^*$ and $2/d^*$ (p. 167) and is written as $r/d_0\prod_1^t\sigma_i\prod_u^kp_j$; adding $1/d_0$ back, $a-\sum_{d\le d^-(d^*)}1/d=(1/d_0)\,r^*/\prod_1^t\sigma_i\prod_u^kp_j$ with $r^*$ between $\prod_1^t\sigma_i\prod_u^kp_j$ and twice it (p. 168). Lemma 5 writes $r^*=\sum_1^mf_i$ with distinct divisors $f_i\le\prod_1^t\sigma_i\prod_u^kp_j/2p_k\sigma_t^3\log\sigma_t$ (so printed; Lemma 5 on p. 164 gives $\ge$, which is what bounds the cofactors $d_i=\prod_1^t\sigma_i\prod_u^kp_j/f_i$ by $2p_k\sigma_t^3\log\sigma_t$), so $a=\sum_{d\le d^-(d^*)}1/d+(1/d_0)\sum_1^m\varepsilon_i/d_i$ with largest denominator $d_0d_m\le2p_k\sigma_t^3\log\sigma_t$ (so printed; the bound omits the factor $d_0$, and since $d_0<4\sigma_t$ by Bertrand's postulate, restoring it changes the logarithm of the bound only by $O(\log a)$). Lemma 4 with $a\le\sigma_t<2a$ gives $p_k\le\exp[a(1+3/\log a)]$ for $a\ge e^3$, hence $d_0d_m\le a^4\exp[a(1+3/\log a)]$, so "$a\in N(n)$ provided $a^4\exp[a(1+3/\log a)]\le n$. But this implies that $a\le\log n(1-\frac{5\log_2n}{\log n})$" (p. 168). That implication runs from the condition to the range; the opening sentence needs the converse, which fails: at $a=\log n-5\log\log n$ the logarithm of $a^4\exp[a(1+3/\log a)]$ is $\log n+(3+o(1))\log n/\log\log n$. As printed, the argument reaches $a$ only up to $\log n-(3+o(1))\log n/\log\log n$, enough for $|N(n)|\sim\log n$ but not for the error term $5\log\log n$ of Theorem 1; what the 1998 Corrigendum changes is unknown here. "Thus $|N(n)|/\log n\ge(1-5\log_2n/\log n)$. Hence $|N(n)|\sim\log n$" (p. 169). The printed text does not remark on the distinctness of the two families of denominators, $d\le d^-(d^*)$ and $d_0d_i$; nothing is claimed about it here.
• References (p. 169, page image), five items: Erdős and Graham, Old and New Problems and Results in Combinatorial Number Theory, pp. 30--44 (1980); Guy, Unsolved Problems in Number Theory, 2nd ed. (1991); Tenenbaum, Introduction to Analytic Number Theory and Probabilistic Number Theory, English ed. (1995); and the author's papers Length and denominators of Egyptian fractions, II (J. Number Theory 28, 1988) and On number of integers representable as sums of unit fractions (Canad. Math. Bull. 33, 1990).

Compiled scope

The paper is compiled at statement depth for the result the citing problems consume: Theorem 1 (p. 162), with the initial-segment form its proof states (p. 167), read on the page images and paged on theorem_1. Lemma 4 (p. 164), which the problem pages cite, has its own page. The lemmas are recorded as statements read on the page image; the proofs were read for structure only, and nothing is independently reviewed. The 1998 Corrigendum is not held.

Bears on. #309: Theorem 1 (printed p. 162, PDF p. 1) gives the lower bound the site's commentary credits to the paper: "There exists a constant $n_0$ such that for all $n>n_0$ $(1-\frac{5\log_2n}{\log n})\le\frac{|N(n)|}{\log n}<(1+\frac1{\log n})$", where $N(n)$ contains $0$ (every $\varepsilon_i=0$), so the problem's count $F(N)$ of positive integers is $|N(N)|-1$; the lower bound reads $F(N)\ge\log N-5\log\log N-1$, the site's $\log N-O(\log\log N)$, and with the trivial upper bound gives $F(N)\sim\log N$, so $F(N)$ is not $o(\log N)$. The upper bound is the trivial $|N(n)|\le\log n+1$ recalled on the same page. The 1998 Corrigendum is not held, so the theorem is quoted as printed in 1997; Croot's 1999 introduction states the range $\{n:1\le n\le\log x-5\log\log x\}\subseteq N(x)$ but credits it to the Corrigendum (his [6]), so it does not confirm the 1997 printing, whose last step (p. 168) does not reach that range (see Contents); the printed argument still gives $|N(n)|\sim\log n$, so the disproof stands. Lemma 4 (printed p. 164), used in the proof of Theorem 1 (p. 168), is quoted as Theorem 1 of the author's 1990 paper, whose count bound $|N(n)|\ge(\frac12-\varepsilon(n))\log n$ the introduction recalls; the lemma bounds $p_k$ and is not itself a count of $N(n)$. #308: the opening sentence of the proof (printed p. 167, PDF p. 6), "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", states the initial-segment form: for large $N$, $\{1,\ldots,\lfloor\log N-5\log\log N\rfloor\}\subseteq N(N)$, which would put the smallest integer not in $N(N)$ above $\log N-5\log\log N$. The printed last step (p. 168) reaches only the integers up to $\log N-(3+o(1))\log N/\log\log N$ (see Contents), and Croot's Main Theorem takes its small integers from "the main result in [5] (and [6])", this paper with its Corrigendum (typescript p. 12). The introduction (p. 162) records Erdős's question for the size of the smallest integer not in $N(n)$, the problem's first question, citing Guy's book.

Results.

• Theorem 1 (p. 162): for all $n>n_0$, $(1-\frac{5\log_2n}{\log n})\log n\le|N(n)|<(1+\frac1{\log n})\log n$; its proof opens by stating that every positive integer $a\le\log n(1-5\log_2n/\log n)$ lies in $N(n)$ (p. 167), a range its printed last step (p. 168) does not reach.
• Lemma 4 (p. 164): for $a$ strictly between the sums of $1/d$ over the divisors $d\le p_k$ and $d\le p_{k+1}$ of $\prod_1^t\sigma_i\prod_u^kp_j$, $p_k\le\exp(\frac{a-1}{1-1/\log s_t-3/\log^2s_t})$, quoted as Theorem 1 of the author's 1990 paper.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.