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Let $$Q(x,y) = ax^2 + bxy + cy^2, \quad a,b,c \in \mathbb{Z}$$ be a binary quadratic form. We say an integer $n$ is representable by $Q$ if $n = Q(x,y)$ for some $x,y \in \mathbb{Z}$. A theorem due to Fermat shows that a prime $p$ is representable by $x^2 + y^2$ if and only if $p = 2$ or $p \equiv 1 \pmod 4$. Fermat also proved similar theorems about the quadratic forms $x^2 + 2y^2$ and $x^2 + 3y^2$.

Given a quadratic form $Q$, one might ask if there exists an theorem analogous to those of Fermat. More precisely, does there exist a modulus $M \in \mathbb{N}$ and congruence classes $a_1,\ldots,a_k$ such that for all but finitely many primes $p$, we have that $p$ is representable by $Q$ if and only if $$p \equiv a_1,\ldots,a_k \pmod M.$$ We say a natural number $N \in \mathbb{N}$ is convenient if the above holds for the quadratic form $Q_N(x,y) = x^2 + Ny^2$. Gauss proved that the $65$ numbers $$1,2,3,4,7,5, 6, 8, 9, 10, 12, 13, 15, 16, 18, 22, 25, 28, 37, 58, 21, 24, 30, 33, 40, 42, 45, 48, 57, 60, 70, 72, 78, 85, 88, 93, 4 102, 112, 130, 133, 177, 190, 232, 253 105, 120, 165, 168, 210, 240, 273, 280, 312, 330, 345, 357, 8 385, 408, 462, 520, 760 840, 1320, 1365, 1848$$ are convenient, and Weinberger showed that there are at most two additional convenient numbers.

My question is what is known about the general case? For instance, is there a finite list of quadratic forms for which a Fermat-type theorem holds?

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