
and

,

are integers such that

–

is a square-free positive integer P. For example P could be 3*5, but not 3^2*5. Let

be the number of pairs of integers

such that

. Show that

is finite and that

for every positive integer

.
Original Statement:
Let

be given integers

where

are (distinct) prime numbers. Let

denote the number of pairs of integers

for which

Prove that

is finite and

for every integer

Note that the "

" in

and the "

" in

do not have to be the same.
%V0
$a > 0$ and $b$, $c$ are integers such that $ac$ – $b^2$ is a square-free positive integer P. For example P could be 3*5, but not 3^2*5. Let $f(n)$ be the number of pairs of integers $d, e$ such that $ad^2 + 2bde + ce^2= n$. Show that$f(n)$ is finite and that $f(n) = f(P^{k}n)$ for every positive integer $k$.
Original Statement:
Let $a,b,c$ be given integers $a > 0,$ $ac-b^2 = P = P_1 \cdots P_n$ where $P_1 \cdots P_n$ 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 $k \geq 0.$ Note that the "$n$" in $P_N$ and the "$n$" in $M(n)$ do not have to be the same.