How many ways can a quadratic form represent a prime?

Question

Given $a,b,c,p\in\Bbb N$ with $b^2-4ac<0$ and $p$ is a prime with $\bigg(\frac{b^2-4ac}p\bigg)=1$, how many solutions $(x,y)\in\Bbb Z^2$ are there to $$ax^2+bxy+cy^2=p?$$

Answer 1

you are missing some concepts. If $p$ does not divide the discriminant $\Delta = b^2 - 4 a c$ and $(\Delta|p) = 1,$ also $\Delta < 0,$ then at least one primitive form of that discriminant does, in fact, represent the prime. One may, further, demand that the form be Gauss reduced, so that we can tell by sight whether it is ambiguous, that is improperly equivalent to itself. If so, and if it does represent the prime, then one expects four representations, of very slightly different appearance. For example, if $a u^2 + c v^2 = p,$ then we have four solutions $(u,v); (-u,-v); (u,-v);(-u,v).$ Only in the extremely special cases of $x^2 + y^2$ and $x^2 + xy + y^2$ can we get eight representations of a prime.

On the other hand, consider primes for which $(-44|p) = 1.$ These include $3,5.$ But these have no expression as $x^2 + 11 y^2$ Instead, they are represented by the other forms in the class group, a pair of opposite forms, $3 x^2 + 2 xy + 4 y^2$ and $3 x^2 - 2 xy + 4 y^2$