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Statement

Setting (p. 311). For an integer $n\ge2$, $P(n)$ is the largest prime factor of $n$.

Theorem 1 (pp. 311--312, quoted). "For each $\epsilon>0$, there is a $\delta>0$ such that for sufficiently large $x$, the number of $n\le x$ with

$$x^{-\delta}<P(n)/P(n+1)<x^{\delta} \tag{1}$$

is less than $\epsilon x$."

So $P(n)$ and $P(n+1)$ are usually far apart on the scale $x^{\delta}$. The theorem says nothing about which of the two is larger.

Source. P. Erdős, C. Pomerance, On the largest prime factors of $n$ and $n+1$, Aequationes Math. 17 (1978), 311--321, read in the edition named on the source card: the statement on pp. 311--312, the proof in §3 (pp. 314--316).

Read depth. Claims checked: the statement was read clause by clause on the printed pages. The proof was read for the pointer below and not checked step by step; nothing here is independently reviewed.

Proof pointer

§3 (pp. 314--316). By Dickman's theorem (the paper's Theorem A, p. 311) one discards, for a small $\delta_0=\delta_0(\epsilon)$, the $n$ with $P(n)<x^{\delta_0}$ or $x^{1/2-\delta_0}\le P(n)<x^{1/2+\delta_0}$. When $P(n)<x^{1/2-\delta_0}$, counting pairs of primes $p=P(n)$, $q=P(n+1)$ with $px^{-\delta}<q<px^{\delta}$ and applying Lemmas 1 and 2 (p. 313) bounds the count by a quantity of order $\delta x/\delta_0$. When $P(n)\ge x^{1/2+\delta_0}$, writing $n=aP(n)$ and $n+1=bP(n+1)$, Brun's sieve (Halberstam and Richert, Sieve Methods, Theorem 2.3) bounds the $n$ for each pair $a,b$, and Landau's asymptotic for $\sum_{n\le x}1/\varphi(n)$ sums the bound. Choosing $\delta$ small in terms of $\epsilon$ and $\delta_0$ (conditions (4) and (7)) finishes.

Depends on. Theorem A (Dickman) and Lemmas 1 and 2 of the paper (pp. 311, 313); Brun's sieve in the form of Halberstam and Richert; Landau's estimate for $\sum 1/\varphi(n)$. None is recorded here.

Bears on

• Problem 371: the theorem does not order $P(n)$ and $P(n+1)$ and gives no density for either ordering. The paper's positive lower density for each ordering (corollary, p. 319) uses an argument similar to case (i) of this proof.
• With Theorem 2 it gives that the Aaron numbers, the $n$ with $f(n)=f(n+1)$, have density $0$ (p. 312); no problem page of this corpus asks this.