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Statement

The paper's equation (1) is

$$\frac an=\frac1x+\frac1y+\frac1z$$

in positive integers $x,y,z$, repetition allowed (p. 193); Schinzel's conjecture is stated "for every $a>0$", the congruences modulo $a$ in (3) take $a$ to be a positive integer, and the paper takes $a>3$ throughout because (1) always has a solution when $a=1$, $2$ or $3$. $E_a(N)$ denotes the number of natural numbers $n\le N$ for which (1) has no solution (Definition, p. 193).

Theorem (p. 193, the paper's only theorem, unnumbered). For each fixed $a$,

$$E_a(N)\ll N\exp\Bigl\{-\frac{(\log N)^{2/3}}{C(a)}\Bigr\},$$

the paper's (2), "where $C(a)$ is a positive number depending at most on $a$" (p. 193). The implied constant is not made explicit, and the closing lines (p. 198) obtain the bound for $N>C_7(a)$, so both constants depend on $a$ alone. The paper adds that the theorem implies that almost every $n$ has a representation in the form (1).

For $a=4$ this bounds the exceptions to the Erdős--Straus conjecture: the number of $n\le N$ with $4/n$ not a sum of three unit fractions is at most a constant times $N\exp(-c(\log N)^{2/3})$ with $c=1/C(4)$, the form in which the site and the later literature quote it. Since this bound is $o(N/\log N)$ while there are about $N/\log N$ primes up to $N$, almost every prime $p$ has a representation of $4/p$ (a consequence drawn here, not printed in the paper).

Source. R. C. Vaughan, On a problem of Erdős, Straus and Schinzel, Mathematika 17 (1970), 193--198; the definition and the theorem on printed p. 193 (PDF p. 1 of the publisher's PDF), the sieve inequality (5) on p. 194 (PDF p. 2), the closing estimate on p. 198 (PDF p. 6), read on the page images; the text layer garbles the displays, including the theorem's. The artifact is identified in the source digest.

Read depth. Claims checked: the definition, the theorem and the sentence drawing the almost-every consequence were read clause by clause on the page image on 2026-09-22. The proof (pp. 193--198, the whole paper) was read on the page images for structure as recorded below; Lemmas 1 and 2 (pp. 193--194), a line and a paragraph, were followed; the proof of Lemma 7 (pp. 195--197) and the Rankin argument (pp. 197--198) were not checked line by line. Nothing here is independently reviewed.

Proof pointer

Pages 193--198, four steps.

1. Explicit solutions (Lemma 1, p. 193). If $rn+s\equiv0\pmod{arst-1}$ for positive integers $r,s,t$, then (1) has a solution: writing $rn+s+q=arstq$, the triple $x=stq$, $y=nrtq$, $z=nrst$ has $1/x+1/y+1/z=(nr+s+q)/(nrstq)=a/n$ (followed here). For $a=4$ the paper refers to Chapter 30, § 1 of Mordell's Diophantine equations for similar solutions.
2. Residue classes modulo a prime (Lemma 2, p. 194). For a prime $p\equiv-1\pmod a$, the triples $(r,s,t)$ with $arst=p+1$, $t$ squarefree and $s\le((p+1)/(at))^{1/2}$ give pairwise distinct classes $n\equiv-s/r\pmod p$, each of which makes (1) soluble by Lemma 1 with $arst-1=p$. Their number is at least $f(p)=[f_1(p)]$, where $f_1(p)=\frac12\sum_{t\mid(p+1)/a}|\mu(t)|\,d\bigl(\frac{p+1}{at}\bigr)$ for $p\equiv-1\pmod a$ and $f_1(p)=0$ otherwise (the paper's (3) and (4), p. 193).
3. The large sieve (Lemma 3, p. 194, a special case of the corollary to Theorem 2 of Montgomery's 1968 note). Removing $f(p)$ classes modulo each prime $p\le\sqrt N$ from $\{1,\ldots,N\}$ leaves at most $4N/S$ integers, with $S=\sum_{s\le\sqrt N}\mu^2(s)\prod_{p\mid s}f(p)/(p-f(p))$; every $n\le N$ without a representation survives the sieve, so $E_a(N)\le4N/S$ (the paper's (5) and (6)).
4. Estimating $S$ (pp. 194--198). Lemma 7 (p. 195) gives $(\log X)^2/C_1(a)<\sum_{p\le X}f(p)/p<C_2(\log X)^2$ for large $X$; the lower bound uses the Bombieri--Vinogradov theorem (Lemma 4, quoted from Davenport's Multiplicative number theory) through Lemma 5, and the upper bound the Brun--Titchmarsh inequality (Lemma 6, quoted from Prachar). Rankin's method (pp. 197--198) compares $S\ge G(\sqrt N,X)$, the sum over squarefree $s\le\sqrt N$ composed of primes $p\le X$, with the full product $G(\infty,X)=\prod_{p\le X}(1-f(p)/p)^{-1}$; with $X=\exp\{((\log N)/(4eC_2))^{1/3}\}$ the tail is less than half of the product, so $S\gg\exp\{(\log N)^{2/3}/C_6(a)\}$ for $N>C_7(a)$, and (5) gives the theorem.

Dependencies

Within the paper: Lemmas 1--7. Outside it, none held here: Montgomery, A note on the large sieve, J. London Math. Soc. 43 (1968), 93--98 (the paper's [1]); Bombieri's theorem in the form of Theorem 1 of Chapter 24 of Davenport, Multiplicative number theory (1967), the paper's [2], due to Bombieri, On the large sieve, Mathematika 12 (1965), 201--225 (its [11]); the Brun--Titchmarsh inequality as Satz 4.1, Kapitel II of Prachar, Primzahlverteilung (1957), the paper's [10]; Rankin's method, used without a citation.

Bears on

• Problem 242: with $a=4$ the theorem is the bound $N\exp(-c(\log N)^{2/3})$ on the number of $n\le N$ without a representation that the site's commentary attributes to Vaughan, now first-hand; it shows the conjecture holds for almost every $n$ and for almost every prime, and says nothing about whether any exception exists. The paper's convention allows repeated denominators; the page's Formulation converts a representation into one with three distinct terms. The uniform version in the numerator is Theorem 1.3 of Pomerance and Weingartner, whose proof its authors describe as largely derivative of this one.