---

Statement

With $S(N)$ and $\log_j$ as on the Theorem 2 page:

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

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

For $r\ge2$, since $\prod_{j=1}^r\log_jN=\log N\cdot\log_2N\cdot\prod_{j=3}^r\log_jN$, this reads

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

the form printed in the introduction (p. 598; at $r=1$ the theorem is the stronger $\log S(N)\le N/\log_2N$ of Lemma 8, which implies it), quoted as "Theorem 3 of [2]" in Corollary 4 of the 1975 paper, and displayed on the site's pages for Problems 320 and 321.

Source. M. N. Bleicher and P. Erdős, Denominators of Egyptian fractions II, Illinois J. Math. 20 (1976), 598--613; Theorem 3 on printed p. 610 (PDF p. 13), proof pp. 610--612; the cases $r=1,2$ are Lemma 8 (p. 607). Read on the page image of p. 610.

Read depth. Claims checked: the statement was read clause by clause on the page image. The proof was read for its structure (below) and is not verified here.

Proof pointer

Induction on $r$, the cases $r=1,2$ being Lemma 8. With $Q=N/\log N$ and $Q'=N/\log_2N$, the integers $k\le N$ are split into $Z_1$, those with a prime factor $p$ in $(Q,N)$, and $Z_2$, the rest, so that $S(N)\le S_1(N)S_2(N)$ with $S_i$ the count of distinct sums over $Z_i$. Writing $S_i^*=\log S_i$: every sum over $Z_2$ has denominator dividing $\mathrm{lcm}(Z_2)$, which with Rosser--Schoenfeld's [4, Theorem 4] gives $S_2^*(N)\le2N/\log N$ (display (4), p. 610); the sums over $Z_1$ are grouped by the prime $p$ and bounded by induction, using Lemma 9 (p. 610) for $\sum_{Q<p\le Q'}\frac1{p\log(N/p)}$ (pp. 610--612, read for structure only). Bloom's exposition of 1 September 2026 on the site's Problem 320 page describes the same decomposition as the recursive bound $s(x)\ll x/\log x+\sum_{x/\log x<p\le x}s(x/p)$ for $s=\log S$.

Dependencies

Rosser--Schoenfeld's explicit bounds (the paper's [4]); Lemmas 8 and 9 of the same paper.

Bears on

• Problem 320: the classical upper bound for $\log S(N)$; the 2026 upper bound of the same order as the lower bounds (site-accepted, unrefereed) refines the same recursion, as the problem page records.
• Problem 321: since a set $A\subseteq\{1,\ldots,N\}$ with distinct subset reciprocal sums has $2^{|A|}\le S(N)$, the theorem gives $R(N)\le\frac1{\log2}\frac{N\log_rN}{\log N}\prod_{j=3}^r\log_jN$ for $\log_{2r}N\ge1$, the upper bound the site prints.