Let $p^n$ be a prime power and, for integers $a,b,c$, let $Q(x,y)=ax^2+bxy+cy^2$ where $p\nmid b^2-4ac$. Define $$N(k,m):=|\{(i,j)\in\{0,\ldots,m-1\}^2: \gcd(i,j,m)=1, Q(i,j)\equiv k\pmod{m}\}|.$$

I wish to prove that, for all $k$, $N(k,p^n)=N(k,p)p^{n-1}$.

For example, the number of solutions of $2x^2-2xy+3y^2=k$ satisfy $N(k, 3)=4,2,2,4,2,2,4,2,2,\ldots$, while $N(k,27)=36,18,18,36,18,18,36,18,18,\ldots$ ($k=0,1,\ldots$).

This is a followup question to question number 424856.

Let $p^n$ be a prime power and, for integers $a,b,c$, let $Q(x,y)=ax^2+bxy+cy^2$ where $p\nmid b^2-4ac$. Define $$N(k,m):=|\{(i,j)\in\{0,\ldots,m-1\}^2: \gcd(i,j,m)=1, Q(i,j)\equiv k\pmod{m}\}|.$$

I wish to prove that, for all $k$, $N(k,p^n)=N(k,p)p^{n-1}$.

For example, the number of solutions of $2x^2-2xy+3y^2=k$ satisfy $N(k, 3)=4,2,2,4,2,2,4,2,2,\ldots$, while $N(k,27)=36,18,18,36,18,18,36,18,18,\ldots$ ($k=0,1,\ldots$).

This is a followup question to a previous question here.