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Statement

Setting (p. 143). $f(x)$ is a polynomial with rational integer coefficients, $n$ runs through $0,1,2,3,\ldots$, and $P_n$ is the largest prime factor of $f(n)$. Equation (3) of the paper is

$$\lim_{n\to\infty}P_n=\infty .$$

Satz I (p. 144, quoted). "Ist $f(x)$ das Produkt zweier wesentlich verschiedenen, d. h. nicht nur um eine multiplikative Konstante verschiedenen rationalen Linearfaktoren, so gilt (3)."

In English: if $f(x)$ is the product of two rational linear factors that are essentially different, that is, that do not differ merely by a constant factor, then $P_n\to\infty$. Equivalently, for every finite set of primes only finitely many $n$ make $f(n)$ a product of primes from that set (the proof is run in this form).

The paper presents Satz I as Thue's result: Thue, using his theorem on Diophantine equations, generalized Størmer's theorem that (3) holds for $x(x+1)$ and $x(x+2)$ (pp. 143-144).

Sharpness (p. 144). The hypothesis cannot be dropped. If the two linear factors differ only by a constant factor, or more generally $f(x)=c(ax+b)^m$, then (3) fails: setting aside $b=0$, one may assume $(a,b)=1$, $a\ge1$, $b>a$, and for $n=\frac{b}{a}\bigl(b^{\varphi(a)\nu}-1\bigr)$, $\nu=1,2,3,\ldots$, the number $P_n$ is the largest prime dividing $bc$, so it takes the same value for infinitely many $n$.

Source. Georg Pólya, Zur arithmetischen Untersuchung der Polynome, Mathematische Zeitschrift 1 (1918), 143-148, doi:10.1007/BF01203608: the setting on p. 143, Satz I and the sharpness remark on p. 144, the proof on p. 145. The edition read is identified on the source card.

Read depth. Claims checked: the statement, its setting and the sharpness remark were read clause by clause on the printed pages. The proof was followed for structure and not verified. Nothing here is independently reviewed.

Proof pointer

P. 145 (end of § 2), following Thue (the paper's footnote 6 cites Thue's first memoir, Satz 12, p. 30). Write the factors as $an+b$ and $cn+d$ with $b/a\ne d/c$, display (6). If (3) failed, there would be finitely many primes $p_1,\ldots,p_l$ and infinitely many $n$ with $(an+b)(cn+d)$ divisible by no other prime. Reducing the exponents modulo $3$ writes $an+b=p_1^{r_1}\cdots p_l^{r_l}x^3$ and $cn+d=p_1^{s_1}\cdots p_l^{s_l}y^3$ with exponents $0$, $1$ or $2$, so only $3^{2l}$ exponent systems occur. Each such $n$ gives a solution of $a\,p_1^{s_1}\cdots p_l^{s_l}y^3-c\,p_1^{r_1}\cdots p_l^{r_l}x^3=ad-bc$, whose right side is nonzero by (6) and whose left side is not the cube of a linear form in $x,y$. Infinitely many such $n$ would contradict the second form of Thue's theorem.

Dependencies

Thue's theorem as the paper states it in § 2 (pp. 144-145), in its second form: if $c\ne0$ and $F(x,y)=c$ has infinitely many integer solutions, then the binary form $F$ is, up to a constant factor, a power of a linear form or of an indefinite quadratic form. The paper cites it from Thue's papers and does not prove it.

Bears on

• Problem 368: the case $f(x)=x(x+1)$ gives that the largest prime factor of $n(n+1)$ tends to infinity, with no rate. The problem asks how large that prime is, so this is a partial result. The claim page records the relation.
• Problem 891: the paper's equivalent form of Satz I, the growing gaps between integers built from a fixed set of primes, is on the page for equation (14), which states its relation to the problem.