Prove Proposition (8.9).
Let's remind ourselves of Proposition (8.9).
A positive integer which looks like $4^n (8m + 7)$ cannot be expressed as a sum of three squares.
For the purpose of contradiction, let's assume $4^n(8m+7)$ can be expressed as a sum of three squares.
$$ 4^n(8m+7) = a^2 + b^2 + c^2 + d^2 $$
Noting that $4^n = (2^n)^2$, we have
$$ (2^n)^2(8m+7) = a^2 + b^2 + c^2 + d^2 $$
That is, $2^n$ is a factor of each $a, b, c, d$. This means $\frac{a}{2^n}, \frac{b}{2^n}$, etc are whole numbers.
And so
$$ 8m+7 = (\frac{a}{2^n})^2 + (\frac{b}{2^n})^2 + (\frac{c}{2^n})^2 + (\frac{d}{2^n})^2 $$
This means a natural number of the form $8m+7$ can be expressed as a sum of four squares. However, from the last excericse (8.2).11 we know that $8m+7$ can not be expressed as a sum of four squares.
This contradiction tells us the assumption $4^n(8m+7)$ can be expressed as a sum of three squares is false, proving the Proposition (8.9).