Question 7
(GEO 3) Let $a,b,c$ be given integers $a\gt 0$, $ac-b^2=P=P_1\cdots P_m$ where $P_1,\ldots,P_m$ are (distinct) prime numbers. Let $M(n)$ denote the number of pairs of integers $(x,y)$ for which
$$ax^2+2bxy+cy^2=n.$$
Prove that $M(n)$ is finite and $M(n)=M(P^k\cdot n)$ for every integer $k\ge0$.
$$ax^2+2bxy+cy^2=n.$$
Prove that $M(n)$ is finite and $M(n)=M(P^k\cdot n)$ for every integer $k\ge0$.