= Reduced forms of discriminant minus thirty-two
{title2=$[1,0,8],\ [2,0,4],\ [3,2,3]$}
The reduced-form bound $3a^2\le32$ leaves $a=1,2,3$. Checking $c=(b^2+32)/(4a)$ and the boundary sign convention gives precisely the displayed forms. The middle one is imprimitive and represents the prime two. Odd represented primes are $1$ or $3$ modulo eight by the <discriminant criterion for prime representation by a binary quadratic form>. The first form represents only $1$ modulo eight among odd integers, and the last only $3$, so a prime is represented by $3x^2+2xy+3y^2$ exactly when it is $3$ modulo eight.
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