Bleicher 1976 denominators egyptian fractions ii

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theorem_2: The 1976 lower bound for the number S(N) of distinct subsums of the first N unit fractions, valid whenever the 2r-fold iterated logarithm of N is at least one, with constant one over e.

theorem_3: The 1976 upper bound for the number S(N) of distinct subsums of the first N unit fractions, valid whenever the 2r-fold iterated logarithm of N is at least one; the classical upper bound for Problems 320 and 321.

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M. N. Bleicher and P. Erdős, Denominators of Egyptian fractions II, Illinois J. Math. 20 (1976), 598--613. Received July 5, 1974; revised January 27, 1976. Part I is Denominators of Egyptian fractions, J. Number Theory 8 (1976), 157--168, filed separately.

The copy read for this card is a scan of the sixteen printed pages with an OCR text layer (physical PDF p. $n$ is printed p. $597+n$). The text layer reads the prose but garbles the displayed formulas, so the statements below were checked on the page images of pp. 598, 602, 603, 610 and 612. Provenance: downloaded in September 2026 from the Erdős archive at users.renyi.hu (archive path 1976-10.pdf; the exact URL was not recorded); 1,077,732 bytes. No copyright or license line is printed on pp. 598--599 or 612--613 of the scan; the hosting archive, users.renyi.hu, has a site footer that speaks for the site, not the paper (https://users.renyi.hu/~p_erdos/, read: "(C) 2005-2007 All rights reserved. All material on this site is for scientifics [sic] purposes only."); the Crossref record for DOI 10.1215/ijm/1256049650 (read 2026-10-02) names Duke University Press as publisher and no license, and the publisher's article page on Project Euclid could not be read on 2026-10-02 (bot-blocked on two URL forms); the term is unstated.

Read status: claims checked. Theorems 1, 2, 3 and 4 were read clause by clause on the page images and Lemma 5 in the text layer; no proof was checked. Result pages: theorem_2, theorem_3.

Contents

Definitions (p. 598): a fraction $a/N$ is in Egyptian form when $a/N=1/n_1+\cdots+1/n_k$ with $0<n_1<\cdots<n_k$; $D(a,N)$ is the least possible $n_k$ and $D(N)=\max\{D(a,N):0<a<N\}$. $S(N)$ (Definition, p. 603) is the number of distinct values of $\sum_{k=1}^N\varepsilon_k/k$ with $\varepsilon_k\in\{0,1\}$. Here $\log_1x=\log x$ and $\log_jx=\log(\log_{j-1}x)$.

• Introduction (p. 598): recalls part I's bounds as $D(P)\ge P\log P$ for primes and $D(N)\le KN(\log N)^4$, announces $D(N)\le(1+\varepsilon)N(\log N)^2$ for large $N$ (Theorem 1), the lower bound of Theorem 4 at primes, and, whenever $\log_{2r}N\ge1$,

$$\frac{\alpha N}{\log N}\prod_{j=3}^{r}\log_jN\le\log S(N) \le\frac{N\log_rN}{\log N}\prod_{j=3}^{r}\log_jN$$

for some $\alpha\ge1/e$; on the upper bound for $D(N)$ the authors write: "We conjecture that the exponent 2 can be replaced by $(1+\delta)$ for $\delta>0$."

• Theorem 1 (p. 602): for every $N$, $D(N)\le\lambda^3(N)N(\ln N)^2$, where $2/\log2\ge\lambda(N)\ge1$ and $\lambda(N)\to1$. The proof writes $a/N$ over $\Pi_k=\prod_{i\le k}p_i$ and uses Lemma 4 (a sum-of-divisors representation via the Cauchy--Davenport theorem, pp. 601--602).
• Lemma 5 (p. 603): $S(N)\ge2^{N/\log N}$ for all $N\ge3$.
• Theorem 2 (p. 603): for each $r\ge1$, every $N$ with $\log_{2r}N\ge1$ satisfies

$$S(N)\ge\exp\Big(\alpha\cdot\frac{N}{\log N}\prod_{j=3}^{r}\log_jN\Big),$$

with the constant $\alpha=1/e$. Proved by induction on $r$ (pp. 603--607). Result page theorem_2.

• Theorem 3 (p. 610): for $r\ge1$ and $\log_{2r}N\ge1$,

$$S(N)\le\exp\Big(\frac{N\log_rN}{\log^2N\,\log_2N}\prod_{j=1}^{r}\log_jN\Big),$$

which is the upper bound quoted in the introduction after canceling $\log N\cdot\log_2N$ when $r\ge2$ (at $r=1$ it is the stronger $\log S(N)\le N/\log_2N$). The cases $r=1,2$ are Lemma 8 (p. 607); the rest is an induction (pp. 610--612). Result page theorem_3.

• Theorem 4 (p. 612): every prime $P$ with $\log_{2r}P\ge1$ satisfies

$$D(P)\ge\frac{P\cdot\log P\cdot\log_2P}{\log_{r+1}P\prod_{j=4}^{r+1}\log_jP}.$$

The proof (pp. 612--613) follows the proof of part I's Theorem 1 (p. 158), which the print cites as "Theorem 2 of [1]" (part I's published Theorem 2 is its upper bound), with Theorem 3 in place of the earlier bound on $S(N)$.

• Closing remarks (p. 613): heuristic and experimental reasons suggest the order of $D(N)/N$ is largest at primes; this would follow from $D(MN)\le D(M)D(N)$ for coprime $M,N$ together with part I's $D(P^k)\le2D(P)P^{k-1}$; $D(P)/P$ is not monotone.

Compiled scope

Only the statements listed above were checked, on the page images named; the proofs were not read beyond the pointers given. Nothing here is independently reviewed.

Bears on. #293: the Erdős–Graham monograph of 1980 (p. 35) cites this paper, with part I and the 1975 paper on distinct subsums, for the claim $v(k)\gg k!$ on the least integer $v(k)>1$ that occurs in no $k$-term representation of $1$ by distinct unit fractions; the paper's theorems concern $D(N)$ and $S(N)$ only and state nothing about $k$-term representations. #305: the problem asks whether $D(N)\ll N(\log N)^{1+o(1)}$; Theorem 1 (p. 602) is the exponent-2 upper bound $D(N)\le\lambda^3(N)N(\ln N)^2$ that the site and the monograph attribute to part I, and Theorem 4 (p. 612) sharpens part I's lower bound $D(P)\ge P\lceil\log P/\log2\rceil$ at primes (part I's $\log_2$ is the base-2 logarithm, not this card's iterated $\log_2$). #320: the problem asks to estimate $S(N)$, the number of distinct sums $\sum_{n\in A}1/n$ over $A\subseteq\{1,\dots,N\}$; Theorems 2 and 3 bound $\log S(N)$ between $\alpha(N/\log N)\prod_{j=3}^{r}\log_jN$ and $(N\log_rN/\log N)\prod_{j=3}^{r}\log_jN$ whenever $\log_{2r}N\ge1$. #321: a set $A\subseteq\{1,\dots,N\}$ whose subsets have pairwise distinct reciprocal sums satisfies $2^{|A|}\le S(N)$, so Theorem 3 bounds $|A|$ above by $\log S(N)/\log2$; the paper's own statements concern $S(N)$ and $D(N)$ and do not name such sets.

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