14. Let the positive integer $m$ have the standard factorization
$$m=2^{a_{0}} p_{1}^{\alpha_{1}} \cdots p_{s}^{\alpha_{s}}$$

where $p_{1}, \cdots, p_{s}$ are distinct odd primes, $\alpha_{0} \geqslant 0, \alpha_{1} \geqslant 1, \cdots, \alpha_{s} \geqslant 1$, $s \geqslant 1$, or $m=2^{\alpha}, \alpha_{0} \geqslant 0$. Prove that $m$ can be expressed as the sum of two coprime squares
$$m=x^{2}+y^{2}, \quad(x, y)=1$$

if and only if:
(1) In the case $m=2^{x_{0}}$, $\alpha_{0}=0$ or $\alpha_{0}=1$,
(2) In the case $m=2^{\alpha_{0}} p_{1}^{\alpha_{1}} \cdots p_{s}^{\alpha_{s}}$, $\alpha_{0}=0$ or 1, and
$$p_{i} \equiv 1(\bmod 4) \quad i=1, \cdots, s$$

Moreover, in case (1), the corresponding representation as the sum of two coprime squares is 1, and in case (2), the corresponding representation as the sum of two coprime squares is $2^{s-1}$.