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Statement

Setting (§ 4, pp. 147-148). Let $p_1,p_2,\ldots,p_r$ be $r$ given primes, $r\ge2$ (12). Let $a_0,a_1,a_2,\ldots$ (13) be the integers $p_1^{x_1}p_2^{x_2}\cdots p_r^{x_r}$, with $x_1,\ldots,x_r$ running over all systems of nonnegative integers, arranged in increasing order.

Equation (14) (p. 148). The paper states that Satz I is equivalent to

$$\lim_{n\to\infty}\,(a_{n+1}-a_n)=\infty .$$

Its reason (p. 148): if $\liminf_{n\to\infty}(a_{n+1}-a_n)=k$, then (3), $P_n\to\infty$, would fail for the polynomial $x(x+k)$; conversely, Satz I need only be proved for the polynomials $x(x+k)$.

Equations (15) and (16) (p. 148). The same sequence also satisfies

$$\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=1, \qquad \lim_{n\to\infty}\frac{(\log a_n)^r}{n}=1\cdot2\cdots r\,\log p_1\log p_2\cdots\log p_r .$$

The paper says these lie much less deep than (14). It bases (15) on the fact that the linear form $x_1\log p_1+\cdots+x_r\log p_r$ in integer variables vanishes only when all $x_i$ are $0$, and so takes arbitrarily small values; it obtains (16) by counting the lattice points in the closed $r$-dimensional region cut out by $x_1\ge0,\ldots,x_r\ge0$ and $x_1\log p_1+\cdots+x_r\log p_r\le\log a_n$.

Source. Georg Pólya, Zur arithmetischen Untersuchung der Polynome, Mathematische Zeitschrift 1 (1918), 143-148, doi:10.1007/BF01203608: § 4 on pp. 147-148, with (12) on p. 147 and (13)-(16) on p. 148. The edition read is identified on the source card.

Read depth. Claims checked: the setting and equations (14)-(16) were read clause by clause on the printed pages. The paper's arguments for the equivalence and for (15)-(16) are one-sentence indications and were not verified. Nothing here is independently reviewed.

Proof pointer

(14) is proved through Satz I, whose proof is on p. 145 (see its page); the equivalence is indicated on p. 148 as described above. (15) and (16) are indicated on p. 148 only.

Dependencies

Satz I of the same paper.

Bears on

• Problem 891: the paper does not state the problem. Erdős and Selfridge (1967, p. 430), as recorded on their source card, invoke Pólya's theorem on gaps between such integers and report Schinzel's deduction from it that, with possibly finitely many exceptions, among any $p_1\cdots p_{k-1}p_{k+1}$ consecutive integers one has more than $k$ prime factors. The problem asks the same for the shorter length $p_1\cdots p_k$, so that deduction does not settle it.