= Primitive representation criterion at discriminant minus thirty-five
{title2=$(-35/q)=1\text{ for every prime }q\mid n$}
For odd $n$ prime to 35, proper representation by a positive definite <binary quadratic form> of discriminant $-35$ is equivalent to $-35$ being a square modulo every prime divisor of $n$. A primitive representing vector extends to a determinant-one integer matrix, producing a form with leading coefficient $n$ and middle coefficient $b$ satisfying $b^2\equiv-35\pmod{4n}$. Conversely this congruence constructs such a form. Its local roots lift to prime powers and combine by the <Chinese remainder theorem>. Every class reduces to one of $x^2+xy+9y^2$ and $3x^2+xy+3y^2$. The condition is local at each prime divisor; a product <Jacobi symbol> equal to one alone is insufficient.
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