---

Statement

Setting (p. 4, Chapter I). $D$ is a natural number that is not a square and $A$ is a nonzero squarefree integer dividing $2D$. $\mathfrak M(D,A)$ is the set of all pairs of integers $X,Y$ with $X^2-DY^2=A$. The fundamental pair $u,v$ of $\mathfrak M(D,A)$ is the pair of natural numbers with $u^2-Dv^2=A$ and $v$ smallest. $\mathfrak N(D,A)$ is the set of pairs $x,y$ in $\mathfrak M(D,A)$ with $y\ne0$ and every prime factor of $y$ dividing $D$.

Satz 1 (p. 15). If $A\ne1$ and $A\ne-D$, then $\mathfrak N(D,A)$ is either empty, or consists of the four pairs $(\pm u,\pm v)$, or consists of the eight pairs

$$(\pm u,\ \pm v),\qquad \left(\pm\frac{u^3+3uv^2D}{|A|},\ \pm\frac{3u^2v+v^3D}{|A|}\right),$$

where $u,v$ is the fundamental pair of $\mathfrak M(D,A)$ and the signs are taken in all combinations.

The excluded cases (p. 5). For $A=1$ the paper quotes Størmer's theorem (Videnskabsselskabets Skrifter, 1897): $\mathfrak N(D,1)$ is either empty or consists of $x=\pm u$, $y=\pm v$, where $u,v$ is the fundamental pair of $u^2-Dv^2=1$. For $A=-D$ it shows that $\mathfrak N(D,-D)$ consists only of $x=0$, $y=\pm1$. It adds that $\mathfrak N(D,D)$ is empty for $D\ne2$ and consists only of $x=\pm2$, $y=\pm1$ for $D=2$.

Singular pairs (§ 11, p. 16). The paper calls the pair $D,A$ singular when $\mathfrak N(D,A)$ is the eight-pair set of Satz 1, and regular otherwise. From the formulas of § 11 it derives in § 12 (pp. 17–19; the three statements below are on pp. 18–19):

• every pair $D,-1$ is regular, so for every non-square natural $D$ the set $\mathfrak N(D,-1)$ is empty or consists of the four pairs $(\pm u,\pm v)$; the paper identifies this sharpening of Satz 1 with Størmer's second theorem mentioned on p. 5;
• every pair $D,+2$ is regular;
• there are exactly three singular pairs $D,-2$, namely $D=3,6,123$, with $\mathfrak N(3,-2)=\{(\pm1,\pm1),(\pm5,\pm3)\}$, $\mathfrak N(6,-2)=\{(\pm2,\pm1),(\pm22,\pm9)\}$ and $\mathfrak N(123,-2)=\{(\pm11,\pm1),(\pm2695,\pm243)\}$; for every other $D$ the set $\mathfrak N(D,-2)$ is empty or consists of $(\pm u,\pm v)$ (under the standing assumption $A\ne-D$ of § 2, which excludes $D=2$).

A table of singular pairs for $A_0=\pm1,\pm2$ is printed on p. 20.

Proof pointer

§§ 2–10, pp. 5–15. A result of D. Schepel (Nieuw Archief voor Wiskunde, 1935), quoted on pp. 6–7, describes $\mathfrak M(D,A)$ for $A\ne1$, $A\ne-D$ through the fundamental pair: the solutions are $\pm X_m,\pm Y_m$, where $X_m+Y_m\sqrt D=(u+v\sqrt D)^{2m+1}/|A|^m$. So the question becomes which odd $n=2m+1$ give a $y_n$ with only prime factors dividing $D$; the set of such $n$ is written $\mathfrak n(D,A)$ (p. 8). Since $y_\nu$ divides $y_n$ when $\nu$ divides $n$, the set is closed under odd divisors. Writing $D=D_0D_1$, $A=A_0D_1$, $u=u_0D_1$ with $D_1=(D,A)$ (§ 5, pp. 9–10), a binomial expansion of $y_n$ and congruences modulo $D_1$, $D_0$ and $p^2$ show that every element of $\mathfrak n(D,A)$ is a power of $3$ (§§ 6–7, pp. 10–13) and that $9$ is not an element (§§ 8–9, pp. 13–14). Hence $\mathfrak n(D,A)$ is empty, $\{1\}$ or $\{1,3\}$ (§ 10, pp. 14–15), the three cases of Satz 1.

Read depth

Claims checked: the setting, Satz 1, the excluded cases and the consequences of § 12 were read clause by clause on the page images of the print; the three singular pairs $D,-2$ were checked against $x^2-Dy^2=-2$ here. The proof was followed but not checked step by step. Nothing here is independently reviewed.

Dependencies

External inputs named by the paper: Schepel's description of $\mathfrak M(D,A)$ and Størmer's theorem for $A=1$.

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 1, with Størmer's theorem for $A=1$, is the step that makes the set $M(z)$ of Satz 3 computable, and Satz 3 bears on Problem 368 and Problem 649.