= Solution
Use the <reduction algorithm for a positive definite binary quadratic form>. A determinant-one shear $(x,y)\mapsto(x+ky,y)$ puts the middle coefficient in $[-a,a]$. If the resulting last coefficient is smaller than $a$, the determinant-one substitution $(x,y)\mapsto(-y,x)$ decreases the positive integer leading coefficient. Repeating terminates with a <reduced positive definite binary quadratic form> satisfying $|b|\leq a\leq c$. These substitutions preserve <proper equivalence of binary quadratic forms> and the <discriminant of a binary quadratic form>.
The <enumeration of reduced binary quadratic forms> gives
$$
163=4ac-b^2\geq3a^2,\qquad1\leq a\leq7.
$$
Since $b^2\equiv-163\equiv1\pmod4$, $b$ is odd. For $|b|=1,3,5,7$, the possible values of $(b^2+163)/4$ are respectively $41,43,47,53$. Each is a <prime number> greater than seven. But $ac=(b^2+163)/4$, so its positive divisor $a\leq7$ must be one. Then $|b|\leq a$ forces $b=\pm1$ and $c=41$. The two signs are properly equivalent: substituting $x\mapsto x-y$ in $x^2+xy+41y^2$ gives $x^2-xy+41y^2$. Thus every form in question is properly equivalent to
$$
\boxed{x^2+xy+41y^2.}
$$
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