Mahler 1935 uber den grossten primteiler spezieller

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satz_1: Mahler's theorem that, for D a non-square natural number and A a squarefree divisor of 2D other than 1 and -D, the solutions of X^2 - D Y^2 = A with Y nonzero and every prime factor of Y dividing D are none, the four sign choices of the fundamental pair, or those four together with four more pairs given explicitly by it.

satz_2: Mahler's theorem that for each squarefree A only finitely many non-square natural numbers D with A dividing 2D make the pair D, A singular, so for all large D the solutions of X^2 - D Y^2 = A with Y nonzero and every prime factor of Y dividing D are at most the four sign choices of the fundamental pair.

satz_3: Mahler's theorem that for A_0 one of 1, -1, 2, -2, D_1 squarefree and coprime to A_0 and x_0 coprime to A_0, the number D_1 x_0^2 - A_0 has a prime factor p > z once x_0 > exp(exp((1 + epsilon) z)) and z is large in terms of epsilon; equivalently its largest prime factor exceeds (log log x_0)/(1 + epsilon) for all large x_0.

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Mahler, Kurt, Über den grössten Primteiler spezieller Polynome zweiten Grades. Archiv for Mathematik og Naturvidenskab 41 (1935), no. 6, pp. 3–26. No copyright line is printed on the file's title or last pages, read on the page images because the scan has no text layer, and the hosting archive's page states only "Page copyright CARMA 2012" for the web page itself and no terms for the scanned papers (https://carmamaths.org/resources/mahler/collected.html, read 2026-10-02); the term is unstated.

This German-language monograph (readable scan) studies the Pell-type equation X^2 - D Y^2 = A, for D a natural number that is not a square and A a nonzero squarefree integer dividing 2D, and deduces a lower bound for the largest prime factor of special quadratic polynomials. The theorem stated in the introduction (pp. 3–4) says that if A_0 is one of the four numbers +1, -1, +2, -2, D_1 is squarefree and coprime to A_0, and x_0 is a natural number coprime to A_0, then for every epsilon > 0 and sufficiently large x_0 the value D_1 x_0^2 - A_0 has a prime factor p > (log log x_0)/(1 + epsilon). It is proved as Satz 3 (p. 26) in the equivalent form that D_1 x_0^2 - A_0 has a prime factor p > z whenever x_0 > exp(exp((1 + epsilon) z)) and z exceeds a bound depending on epsilon; Mahler remarks there that the hypotheses that D_1 is squarefree and coprime to A_0 and that x_0 is coprime to A_0 can be dropped. The method extends Størmer's theory of the solutions of x^2 - D y^2 = 1 whose y has all prime factors dividing D, using an explicit description of the solutions of X^2 - D Y^2 = A in terms of the fundamental solution (a lemma of D. Schepel) together with simple estimates from Mahler's earlier note on the largest prime factor of x^2 + 1 and x^2 - 1. Chapter I sets up the sets M(D,A) of all integer solution pairs and N(D,A) of those pairs whose y is nonzero with every prime factor dividing D, records the cases A = 1 (Størmer's theorem) and A = -D and A = D (elementary), notes that Størmer's analogous result for A = -1 will follow as a special case of the general results, and reduces the general problem to A not equal to 1 and not equal to -D (pp. 4–5). Chapter II (pp. 21–26) applies these results to the set M(z) of natural numbers x_0 coprime to A_0 for which D_1 x_0^2 - A_0 is positive with all prime factors below z.

Source: https://carmamaths.org/resources/mahler/collected.html.

Read status. Claims checked: Satz 1 (p. 15) with the cases of pp. 5 and 18–19, Satz 2 (p. 20), Satz 3 (p. 26) and the theorem of the introduction (pp. 3–4) were read clause by clause on the page images of the print. The proofs were followed but not checked step by step.

Bears on. #368: with D_1 = A_0 = 1 and x_0 = 2n + 1 the theorem of pp. 3–4 gives that the largest prime factor of n(n+1) exceeds (log log n)/(1 + epsilon) for every epsilon > 0 and all large n, a deduction recorded on the Satz 3 page; it is a lower bound and does not determine the order the problem asks for. #649: the same specialization gives that P(n(n+1)) tends to infinity, so each fixed pair of primes p, q has at most finitely many n with P(n) = p and P(n+1) = q; this does not decide whether such an n exists, which is what the problem asks.

Results. Satz 1 (p. 15), with Størmer's theorem for A = 1 and the cases A = -D and A = D (p. 5) and the consequences for A = -1, +2 and -2 (pp. 18–19); Satz 2 (p. 20); Satz 3 (p. 26), with the theorem of the introduction (pp. 3–4) and the count of M(z) (p. 23).

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.