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Polya 1918 zur arithmetischen untersuchung der polynome
equation_14: Pólya's reformulation of Satz I: for r at least 2 given primes, the increasing sequence of all integers whose prime factors lie among them has consecutive differences tending to infinity, with the companion facts that consecutive ratios tend to 1 and a lattice-point count of the n-th term.
satz_1: Thue's theorem, as Pólya states and proves it, that if f is a product of two rational linear factors that differ by more than a constant factor, then the largest prime factor of f(n) tends to infinity as n runs through 0, 1, 2, and so on.
satz_2: Pólya's theorem that if f is an irreducible polynomial of second degree with rational integer coefficients, then the largest prime factor of f(n) tends to infinity as n runs through 0, 1, 2, and so on.
Georg Pólya, Zur arithmetischen Untersuchung der Polynome. Mathematische Zeitschrift 1 (1918), 143-148. doi:10.1007/BF01203608. The file prints the digitizing library's terms sheet, not the publisher's notice: "The Goettingen State and University Library provides access to digitized documents strictly for noncommercial educational, research and private purposes" and "Publication and/or broadcast in any form (including electronic) requires prior written permission from the Goettingen State- and University Library.", every other right reserved.
Polya studies P_n, the largest prime factor of f(n), for a polynomial f with rational integer coefficients and n = 0, 1, 2, .... Stormer proved P_n -> infinity for x(x+1), x(x+2) and x^2+1 (p. 143). Satz I (p. 144), which Polya presents as Thue's generalization of the first two cases and proves by Thue's method, gives P_n -> infinity when f is a product of two essentially different rational linear factors; the paper notes that the conclusion fails for c(ax+b)^m. Satz II (p. 144), Polya's own, proves the same for any irreducible quadratic and contains Stormer's x^2+1 case. Both proofs reduce, after fixing a finite prime set and taking exponents mod 3 (mod 3h, h the class number, for the prime ideals in Satz II), to Thue's theorem in its second form (pp. 144-145): if c != 0 and F(x,y) = c has infinitely many integer solutions, the binary form F is, up to a constant factor, a power of a linear form or of an indefinite quadratic form; for the cubic forms used here, that means the cube of a linear form. Section 3 closes (p. 147) with the remark that Thue's results also give P_n -> infinity for x^n-1 and for polynomials in which a certain number of coefficients after the leading one vanish. Section 4 (pp. 147-148) restates Satz I in another form: if a_0 < a_1 < a_2 < ... are all integers of the shape p_1^{x_1}...p_r^{x_r} for a fixed set of r >= 2 primes, then a_{n+1} - a_n -> infinity (equation 14), and this is equivalent to Satz I for the polynomials x(x+k); the same section adds a_{n+1}/a_n -> 1 (15) and the lattice-point count (log a_n)^r/n -> r! log p_1 ... log p_r (16). Paper is in German; the digest is based on the scan read in full, including the closing note that it was received 4 August 1917.
Source: https://gdz.sub.uni-goettingen.de/download/pdf/PPN266833020_0001/LOG_0020.pdf.
Read status. Claims checked: Satz I, Satz II and equations (14)-(16) were read clause by clause on the printed pages. The proofs (pp. 145-148) were followed for structure and not verified.
Bears on. #368: Satz I for 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 it is. #891: the paper does not state the problem; Erdos and Selfridge (1967, p. 430) invoke the gap statement (14) and report Schinzel's deduction from it that, with possibly finitely many exceptions, among any p_1...p_{k-1}p_{k+1} consecutive integers one has more than k prime factors, a longer length than the p_1...p_k the problem asks about.
Results. Satz I (p. 144); Satz II (p. 144); Equation (14) (p. 148, with (15) and (16)). Thue's theorem (pp. 144-145) is cited from Thue and stated on the pages of the two Satze as their dependency.
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