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Source. P. Erdős and J. L. Selfridge, Some problems on the prime factors of consecutive integers, Illinois J. Math. 11 (1967), 428--430 (source card): $\nu(m)$ is defined on p. 428, inequality (1), its derivation and conjecture (2) are on p. 430.

Read depth. Claims checked: the inequality, the derivation and the conjecture were read clause by clause on the printed page. Pólya's theorem is used as the paper states it and was not checked against Pólya's paper.

Statement

Setting. $\nu(m)$ denotes the number of distinct prime factors of $m$ (p. 428); $\pi(k)$, used without definition, is the number of primes not exceeding $k$.

Inequality (1) (p. 430). For a fixed positive integer $k$ (the paper states no range for $k$ here),

$$\liminf_{n\to\infty}\sum_{i=0}^{k-1}\nu(n+i)\ge k+\pi(k)-1. \qquad(1)$$

The paper says that a well-known theorem of Pólya easily implies (1).

Conjecture (2) (p. 430). The authors write that it seems to them that, for every $k$,

$$\liminf_{n\to\infty}\sum_{i=0}^{k-1}\nu(n+i)\le k+\pi(k), \qquad(2)$$

and that perhaps equality always holds in (2). It is posed as a conjecture, not proved.

Proof pointer

Page 430. The product $\prod_{i=0}^{k-1}(n+i)$ is divisible by every prime not exceeding $k$. Pólya's theorem, as the paper states it, says that if $a_1^{(k)}<a_2^{(k)}<\cdots$ are the integers composed only of primes not exceeding $k$, then $a_{i+1}^{(k)}-a_i^{(k)}\to\infty$. Hence for $n$ sufficiently large every one of $n,n+1,\ldots,n+k-1$, with at most one exception, has a prime factor greater than $k$; counting these with the primes up to $k$ gives (1).

Dependencies

Pólya's theorem on the gaps between integers composed of a fixed finite set of primes, which the paper cites by name without a reference; see G. Pólya, Zur arithmetischen Untersuchung der Polynome, Math. Z. 1 (1918), 143--148 (source card).

Bears on

• Problem 890: the problem's first question asks whether $\liminf_n\sum_{0\le i<k}\omega_k(n+i)\le k$, where $\omega_k$ counts only the distinct prime factors exceeding $k$. The paper's (1) and (2) concern the count of all distinct prime factors, so neither is that question as stated. The derivation of (1) shows that for large $n$ at least $k-1$ of $n,\ldots,n+k-1$ have a prime factor greater than $k$, so the liminf in the problem is at least $k-1$; the paper proves no upper bound.