If $n$ is a positive integer then there is a monic irreducible polynomial $f(x)$ such that if $p$ is an odd prime not dividing $n$ nor the discriminant of $f$ then $$ p=x^2+ny^2\iff \left(\frac{-n}{p}\right)=1\quad\text{and}\quad f(x)\equiv 0\mod p\quad\text{has a solution.} $$ This is a simplified version of theorem 9.2 from Cox's book "Primes of the form x^2+ny^2".

Euler was highly interested in the problem of representing primes by quadratic forms $x^2+ny^2$, and he made many impressive discoveries that take the above form. For example, he discovered that $$ p=x^2+27y^2\iff p\equiv 1\mod 3\quad\text{and}\quad x^3-2\equiv 0\mod p\quad\text{has a solution.} $$

My question is: Did Euler conjecture the form of the general solution for arbitrary $n>0$, as stated above?



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