nt.number theory – Set of Quadratic Forms that represents All Primes


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 $x,y, a, b,c in mathbb Z$?


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 $1 leq ileq m-1$.

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 $(m-1)$.


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 ?

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One can answer only the specific case, if he wishes.