---

Statement

Setting. As on the page of Satz 1: $\mathfrak N(D,A)$ is the set of solutions $x,y$ of $x^2-Dy^2=A$ with $y\ne0$ and every prime factor of $y$ dividing $D$, $u,v$ is the fundamental pair, and $D,A$ is singular when $\mathfrak N(D,A)$ has the eight elements of the third case of Satz 1 (p. 16).

Satz 2 (p. 20). For every squarefree number $A$ there are at most finitely many natural numbers $D$ with $A\mid2D$ that are not squares and for which $D,A$ is singular. Consequently, for all sufficiently large $D$ the set $\mathfrak N(D,A)$ is either empty or contains only the pairs $(\pm u,\pm v)$.

Proof pointer

§§ 11–13, pp. 16–20. For a singular pair, § 9 forces $D_1u_0^2=(|A_0|\cdot3^l+A_0)/4$ and $D_0v^2=(|A_0|\cdot3^l-3A_0)/4$ for some natural $l$ (p. 16). For a fixed $A$ the paper lists, by the sign of $A$ and whether $2\mid A$, the possible $A_0$, $D_1$ and the resulting equations for $u_0$ and $\lambda$ (p. 19). Each $\lambda$ yields at most finitely many $D$, and each equation has at most finitely many solutions, by Pólya's theorem that the largest prime factor of $at^2+bt+c$ with $b^2-4ac\ne0$ tends to infinity, so the polynomial is a pure power of $3$ for at most finitely many $t$ (p. 20).

Read depth

Claims checked: Satz 2 and the definition of singular pairs were read clause by clause on the page images of the print. The proof was followed but not checked step by step. Nothing here is independently reviewed.

Dependencies

Satz 1 and § 9 of the same paper; Pólya's theorem on the largest prime factor of a quadratic polynomial, cited without reference.

Source. K. Mahler, Über den grössten Primteiler spezieller Polynome zweiten Grades, Archiv for Mathematik og Naturvidenskab 41 (1935), no. 6, pp. 3–26; the edition read is named on the source card.

Bears on

None directly. Satz 2 bounds the singular pairs met in the count of $M(z)$ on the way to Satz 3, which bears on Problem 368 and Problem 649.