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Statement

Displays (2) and (3), printed p. 124. By a theorem of Rankin, which the paper cites, there is a constant $c_2>0$ such that for every $k$ some consecutive primes $p_r<p_{r+1}$ satisfy

$$k<p_r<p_{r+1}<2k,\qquad p_{r+1}-p_r>c_2\frac{\log k\cdot\log\log k\cdot\log\log\log\log k}{(\log\log\log k)^2}. \tag{2}$$

Every prime factor of the integers strictly between $p_r$ and $p_{r+1}$ is less than $k$, so that, for a constant $c_3>0$,

$$f(k)>c_3\frac{\log k\cdot\log\log k\cdot\log\log\log\log k}{(\log\log\log k)^2}. \tag{3}$$

On printed p. 125 Erdős calls the gap between (1) and (3) extremely large, suggests that $f(k)$ is probably not much larger than the largest gap $p_{r+1}-p_r$ between consecutive primes of $(k,2k)$, and, citing a conjecture of Cramér, writes that "one might guess $f(k)=(1+o(1))(\log k)^2$" (4), adding: "The proof or disproof of (4) seems hopeless, there is of course no real evidence that (4) is true." The page also records that $f(k)$ can be determined in finitely many steps by a theorem of Pólya and Störmer, without an effective bound; that Erdős cannot prove $f$ nondecreasing; and the values "$f(2)=2$, $f(3)=f(4)=3$, $f(5)=f(6)=4$. It seems likely that $f(7)=f(8)=f(9)=f(10)=4$, but $f(13)\ge6$."

Source. P. Erdős, On consecutive integers, Nieuw Arch. Wisk. (3) 3 (1955), 124--128; displays (2)--(3) on printed p. 124 (PDF p. 1) and (4) with the small values on p. 125 (PDF p. 2), read on the page images.

Read depth. Claims checked: the displays and the surrounding sentences were read clause by clause on the page images. The one-line deduction of (3) from (2) is as printed (a composite between consecutive primes of $(k,2k)$ has all its prime factors below $k$); Rankin's theorem is cited, not proved.

Proof pointer

Immediate from Rankin's prime-gap theorem (R. A. Rankin, J. London Math. Soc. 13 (1938), 242--247, the paper's footnote 2): the composites strictly between $p_r$ and $p_{r+1}$ form a run of $k$-smooth integers above $k$ of length $p_{r+1}-p_r-1$.

Dependencies

Rankin's 1938 lower bound for large gaps between consecutive primes, with the gap located in $(k,2k)$ (as stated by Erdős; the location is part of his citation of Rankin and was not checked against Rankin's paper, which is not held).

Bears on

• Problem 961: (3) is the lower bound for $f(k)$ that the problem page records, and (4) is Erdős's guess, on Cramér's conjecture, that $f(k)=(1+o(1))(\log k)^2$.