23.21 Among the following sets, the three elements cannot possibly be the lengths of the three external diagonals of a rectangular box (a diagonal of one of the rectangular faces of the box) is
(A) $\{4,5,6\}$.
(B) $\{4,5,7\}$.
(C) $\{4,6,7\}$.
(D) $\{5,6,7\}$.
(E) $\{5,7,8\}$.
(44th American High School Mathematics Examination, 1993)
[Solution] Let $a, b, c (a \leqslant b \leqslant c)$ represent the lengths of the three edges of the rectangular prism. Let $p, q, r (p \leqslant q \leqslant r)$ represent the lengths of the three face diagonals. By the Pythagorean theorem, we have
$$
p^{2}=a^{2}+b^{2}, \quad q^{2}=a^{2}+c^{2}, \quad r^{2}=b^{2}+c^{2} .
$$

Then
$$
r^{2}=p^{2}+q^{2}-2 a^{2}<p^{2}+q^{2} \text {. }
$$

Upon verification, only $\{4,5,7\}$ does not satisfy this relationship.
Therefore, the answer is $(B)$.