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

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 $$1 leq ileq m-1$$.

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 $$(m-1)$$.

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 he wishes.