Find $n$ that satisfies $\Phi_p(x)=C(x)^2+(-1)^{\frac{n+1}{2}}n\cdot D(x)^2$

Question

Let $p>2$ be a prime number, based on Gauss's formula shows that cyclotomic polynomial $\Phi_p$ can be expressed as $$\Phi_p(x)=A(x)^2+(-1)^{\frac{p+1}{2}}p\cdot B(x)^2$$ where $A,B\in \mathbb{Q}[x]$. My question is whether, $n$ is a positive integer and $C,D\in \mathbb{Q}[x]$ satisfy $$\Phi_p(x)=C(x)^2+(-1)^{\frac{n+1}{2}}n\cdot D(x)^2,$$ what are all the possible values of $n$?