13. Given that $b$ and $c$ are positive integers, and the three quadratic equations $x^{2}+b x+c=k$ (for $k=0,1,2$) all have real roots but no integer roots, find the minimum value of $b+c$.
13. The discriminant of $x^{2}+b x+c=k$ is $D_{k}=b^{2}-4 c+4 k(k=0,1,2)$. According to the problem, $D_{0} \geqslant 0$ and $\sqrt{D_{0}} \cdot \sqrt{D_{1}}=$ $\sqrt{D_{0}+4}, \sqrt{D_{2}}=\sqrt{D_{0}+8}$ are not integers.
$$
\therefore D_{0} \neq 1,4,5,8,9,12 \text {. }
$$
(1) If $D_{0}=b^{2}-4 c=2$ or 6 or 10, then $b$ is even. Let $b=2 m$, then $4 m^{2}-4 c=2$ or 6 or $4, \therefore 4 \mid 2$ or 6 or 4, a contradiction.
(2) If $D_{0}=b^{2}-4 c=3$ or 7 or 11, then $b$ is odd. Let $b=2 m+1$, then
$4 m(m+1)-4 c=2$ or 6 or 10, thus $4 \mid 2$ or 6 or 4, a contradiction.
$\therefore D_{0} \geqslant 13$, i.e., $b^{2}-4 c \geqslant 13$.
Also, the equation $x^{2}+b x+c=k(k=0,1,2)$, when $c=k$, has integer solutions $x=0$ and $x=-6$.
$$
\begin{array}{l}
\therefore c \geqslant 3, b^{2} \geqslant 13+4 c \geqslant 25, \therefore b \geqslant 5 \\
\therefore b+c \geqslant 8 .
\end{array}
$$

Also, $x^{2}+5 x+3=k(k=0,1,2)$ all have no integer roots, $\therefore$ the minimum value of $b+c$ is 8.