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Statement

Printed p. 126. Let $g(k)$ be the least, over all blocks of $k$ consecutive integers each greater than $k$, of the number of members of the block having a prime factor greater than $k$; the paper defines it as "the smallest integer so that among $k$ consecutive integers each greater than $k$ there are at least $g(k)$ of them having prime factors greater than $k$", and notes that the Sylvester--Schur theorem is the statement $g(k)\ge1$.

Theorem 2.

$$g(k)=(1+o(1))\frac{k}{\log k}.$$

A remark on printed p. 128 records that $k=10$, $n=12$ shows $g(k)$ can be smaller than $\pi(2k)-\pi(k)$.

Source. P. Erdős, On consecutive integers, Nieuw Arch. Wisk. (3) 3 (1955), 124--128; the definition of $g(k)$ and Theorem 2 on printed p. 126, the proof on pp. 126--127, the remark on p. 128.

Read depth. Claims checked: the definition and the statement were read clause by clause on the page images. The proof was read for the sketch below; it is not verified.

Proof pointer

The upper bound: the block $k+1,\ldots,2k$ contains $\pi(2k)-\pi(k)=(1+o(1))k/\log k$ primes, and its other members have no prime factor above $k$. For the lower bound it suffices to show that for $n\ge k$ the block $n+1,\ldots,n+k$ (the paper's display (9)) contains $(1+o(1))k/\log k$ members with a prime factor greater than $k$. The paper splits into three ranges (p. 127):

• $k\le n\le2k$: the prime number theorem gives $\pi(n+k)-\pi(n)=(1+o(1))k/\log k$ primes in the block.
• $2k<n\le k^{3/2}$: by the Hoheisel--Ingham estimate (*) the block holds at least $(1+o(1))k/(\tfrac32\log k)$ primes, and, as $n>2k$, at least $(1+o(1))k/(2\cdot\tfrac32\log k)$ further members of the form $2p$ with $p>k$ prime; the two counts sum to $(1+o(1))k/\log k$.
• $n>k^{3/2}$: for $k$ larger than some $k_0$ at least $k/6$ members have a prime factor greater than $k$, by the lemma on prime powers exactly dividing a binomial coefficient and the inequality (6) from the proof of Theorem 1, applied to $\binom{n+k}{k}$.

Dependencies

The prime number theorem; the Hoheisel--Ingham estimate (*) for primes in short intervals (cited to Ingham, Quart. J. Math. 8 (1937), 255--266); the paper's lemma (p. 126) that $p^\alpha\,\|\,\binom{u+t}{t}$ implies $p^\alpha\le u+t$, from Legendre's formula.

Bears on

No problem page of this corpus. The theorem counts the members of a block with a large prime factor, where Problem 961 asks for the least block length guaranteeing one; the problem page does not cite it.