Timeline for The number of solutions to $ax^2+bxy+cy^2\equiv k\pmod{p^{n}}$, $(x,y)\in\{0,\dotsc,p^{n}-1\}^2$

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Jun 20, 2022 at 0:48	review	Close votes
Jul 2, 2022 at 3:09
Jun 20, 2022 at 0:13	history	edited	KConrad	CC BY-SA 4.0	Added hyperlink to earlier question.
Jun 20, 2022 at 0:10	comment	added	KConrad		In your notation $N(k,m)$, all you actually care about are solutions modulo prime powers $p^n$. Moreover, you are interested in solutions $(x,y) \bmod p^n$, where $\gcd(x,y,p^n) = 1$, which is equivalent to $\gcd(x,y,p) = 1$. Such solutions are called primitive because a vector $(a_1,\ldots,a_r) \bmod p^n$ where the $a_i$'s are not all divisible by $p$ is called a primitive vector. Every primitive solution mod $p^n$ reduces to one mod $p^{n-1}$. You want to lift primitive solutions mod $p^{n-1}$ to primitive solutions mod $p^{n}$. Try using Hensel's lemma.
Jun 20, 2022 at 0:00	history	asked	user47804	CC BY-SA 4.0