Three, (This question is worth 20 points)
Find the real number pairs $(p, q)$ such that the inequality $\left|x^{2}+p x+q\right| \leqslant 2$ holds for all $1 \leqslant x \leqslant 5$.

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Three, (This question is worth 20 points)
Find the real number pairs $(p, q)$ such that the inequality $\left|x^{2}+p x+q\right| \leqslant 2$ holds for all $1 \leqslant x \leqslant 5$.
Let $f(x)=x^{2}+p x+q$, the axis of symmetry of the graph of this quadratic function is $x=-\frac{p}{2}$, and it opens upwards.
(1) When $-\frac{P}{2} \leqslant 1$, the interval $[1,5]$ is to the right of the axis of symmetry, so $-2 \leqslant f(1)5$ when, there is $-2 \leqslant f(5)<f(1) \leqslant 2$, i.e.,
$$
\left\{\begin{array}{l}
p<-10, \\
25+5 p+q \geqslant-2, \\
1+p+q \leqslant 2 .
\end{array}\right.
$$

From the second and third equations, we get $1-p \geqslant-5 p-27$, i.e., $p \geqslant-7$, which contradicts the first equation.
In summary, the pair of numbers that satisfy the given conditions is $(-6,7)$.