**A SPECIFIC CASE:**

Any prime number can be classified as either $p \equiv 1 \pmod 3$ or $p \equiv 2 \pmod 3$.

If $p = 3$ or $p = 1 \pmod 3$, then the prime $p$ can be represented by the quadratic form $ x^2 + 3y^2, x,y \in \mathbb Z.$

But what if $p \equiv 2 \pmod 3$?

Is there a quadratic form $ax^2+bxy+cy^2$ such that $p= ax^2+bxy+cy^2, $ when $p \equiv 2 \pmod 3$ where $x,y, a, b,c \in \mathbb Z$?

**GENERAL CASE:**

The general question is, is there a set of quadratic forms which represent all prime numbers?

We will classify the prime numbers, say, by $m$. Any prime is defined by $p \equiv i \pmod m$ where $1 \leq i\leq m-1$.

In above example, $i \in \{1, 2\}, m=3$. Let, the set of quadratic forms is $A$, then the number of elements in $A$ is at-least $(m-1)$.

**QUESTION:**

For a given $m$ can we find a set $A$ such that any prime $p$ can be represented by one of the quadratic form of $A$ ?

If it is possible then how? If there is a condition on $m$, what is it?

Does the question has any relation to the following theorem ?

One can answer only the specific case, if they wish to do so.

**EDIT:**

Is there a finite set of (preferably irreducible) binary quadratic forms such that every prime is represented by **at least one of the forms** in the set?

onlyrepresent the forms of specific residue. In that case there are some trivial examples, like the form $xy$, or $x^2-y^2$. I'm not sure if you can do with only irreducible forms. $\endgroup$ – Wojowu Oct 11 at 23:11