38. How to find all possible expressions of a given odd number as the difference of squares of two integers? How to determine in advance the number of these expressions?
38. Let the number 63 be given. We will decompose it into two factors in all possible ways: $63=1\cdot63=3\cdot21=7\cdot9$. It should be:

$$
63=x^{2}-y^{2}=(x-y)(x+y)
$$

Let's set sequentially:

$$
\begin{array}{lll}
x-y=1, & x-y=3, & x-y=7 \\
x+y=63, & x+y=21, & x+y=9
\end{array}
$$

from which we get:

$$
\begin{array}{lll}
x=32, & x=12, & x=8 \\
y=31, & y=9, & y=1
\end{array}
$$

Therefore,

$$
63=32^{2}-31^{2}=12^{2}-9^{2}=8^{2}-1^{2}
$$

In general: the number of possible representations as the difference of two squares is equal to the number of possible decompositions of the given number into two integer factors.