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Source. Theorem 3, p. 58, with its proof on p. 63, of P. Erdős, On some applications of Brun's method, Acta Univ. Szeged. Sect. Sci. Math. 13 (1949), 57--63, as identified on the source card.

Statement

Write $p_1<p_2<\cdots$ for the primes in increasing order, as the paper does (p. 58).

Theorem 3 (p. 58). Let $c_5$ be any constant and $n$ sufficiently large. Then there are a constant $c_6=c_6(c_5)$ and primes $p_k<p_{k+1}<\cdots<p_{k+r}<n$, with $r=[c_6\log n]$, such that

$$p_{k+i+1}-p_{k+i}>c_5,\qquad i=0,1,\ldots,r-1.$$

The print calls these "$[c_6\log n]$ primes", although the list $p_k,\ldots,p_{k+r}$ has $r+1=[c_6\log n]+1$ members; the display concerns the $r$ gaps between them. The print places the constant $c_6$ after $n$, but it depends only on $c_5$, as the notation $c_6(c_5)$ says and the proof shows.

The paper presents the theorem (p. 58) as a sharpening of Sierpiński's result that $\limsup\min(p_{n+1}-p_n,\,p_n-p_{n-1})=\infty$, that is, that infinitely many primes are isolated on both sides.

Proof pointer

Page 63. By Schnirelmann's sieve bound, the number of $m$ with $p_{m+1}-p_m\le c_5$ and $p_m\le n$ is less than a constant times $n/(\log n)^2$, while $\pi(n)$ exceeds a constant times $n/\log n$; so the gaps of size at most $c_5$ are too few to break every run of $[c_6\log n]+1$ consecutive primes below $n$ when $c_6$ is small in terms of $c_5$. The paper says this gives the theorem immediately.

Read depth

Claims checked: the statement was read clause by clause on the printed p. 58 and the proof on p. 63. Nothing here is independently reviewed.

Bears on

• Problem 238: the problem fixes $c_1,c_2>0$ and asks whether every sufficiently large $x$ has more than $c_1\log x$ consecutive primes $\le x$ with all pairwise differences greater than $c_2$. Theorem 3 with $c_5=c_2$ and $n=x$ gives $[c_6\log x]+1>c_6\log x$ consecutive primes below $x$ whose successive gaps, and hence (as the primes increase) all pairwise differences, exceed $c_2$; this answers the question yes for every pair with $c_1\le c_6(c_2)$. The paper gives no value of $c_6(c_2)$, and the theorem says nothing about larger $c_1$.