Finding the positive integers that can be written in the form $x^2+xy+5y^2$

Question

I'm working on a homework problem that asks which positive integers can be written in the form $x^2+xy+5y^2$. An example was given on how to find all positive integers that can be written as the difference of two squares (i.e. $x^2-y^2$), but it was a proof by cases that doesn't seem to carry over to more generalized versions of the form $ax^2+bx+cy^2$.

Is there any method to approaching these kinds of problems?

Answer 1

See this text. It describes precisely which numbers are represented by a binary quadratic form (Proposition 4.1). In your case the discriminant is $-19$, so a number $n$ is represented by your form iff $-19$ is a square modulo $4n$. You can conclude by using the Gauss reciprocity law.

Answer 2

There have been a few methods mentioned, so I hope you will forgive me if I mention a sophisticated but fun approach.

Let $x^2 + bxy + cy^2$ be the binary quadratic form of discriminant $d = b^2 - 4c < 0$ such that $$ b = \cases{0 \quad d \equiv 2, 3 \pmod{4} \\ 1 \quad d \equiv 1 \pmod{4}}$$

Note that we are in this case since $d = 1 - 20 = -19 \equiv 1 \pmod{4}$. There is a fun result

Thm A prime $p \nmid d$ is represented by our BQF if and only if $p$ splits completely in the Hilbert Class Field of $L = \mathbb{Q}(\sqrt{d})$

For those who have not seen it before, we have the following

Fact: The Hilbert Class Field is an abelian extension of degree $h_L$ (the class number of $L$).

In our case this gives a way to derive the result since $p$ is represented if and only if $p$ splits (completely) in $\mathcal{O}_L$ where $L = \mathbb{Q}(\sqrt{-19})$ (this is because $h_L = 1$). If $p \neq 19$ we see that this is the case if and only if the polynomial $x^2 + 19$ splits mod $p$. This is if and only if $-19$ is a square mod $p$.