Number theory in sequence $x_{n+1}=x_n^3-2x_n^2+2$ $x_1=5, x_{n+1}=x_n^3-2x_n^2+2$
Prove that, there is no prime $p=4k+3(k>1)$ and $p\mid x_n^2-3x_n+3$
I think I can have $p\mid t^2+1$ and then I have QED.
But $p\mid x_n^2-3x_n+3$ means that $p\mid (2x_n-3)^2+3=t^2+3$
What should I do next?
 A: I found this to be a quite interesting (and challenging to solve) question. It's asking about the primes $p$ where
$$p \mid x_n^2 - 3x_n + 3 \tag{1}\label{eq1A}$$
As you showed, multiplying by $4$ gives
$$p \mid 4x_n^2 - 12x_n + 12 = (2x_n - 3)^2 + 3 \tag{2}\label{eq2A}$$
Since $p \neq 3$ (note $x_n \equiv 5 \pmod{72}$ for all $n \ge 1$, with this giving $x_n^2 - 3x_n + 3 \equiv 13 \pmod{72}$), this shows $-3$ is a quadratic residue modulo $p$.
Next, multiplying \eqref{eq1A} by $x_n - 3$ gives
$$\begin{equation}\begin{aligned}
(x_n - 3)(x_n^2 - 3x_n + 3) & = x_n^3 - 3x_n^2 + 3x_n - 3x_n^2 + 9x_n - 9 \\
& = x_n^3 - 6x_n^2 + 12x_n - 8 - 1 \\
& = (x_n - 2)^3 - 1
\end{aligned}\end{equation}\tag{3}\label{eq3A}$$
This means
$$(x_n - 2)^3 \equiv 1 \pmod{p} \tag{4}\label{eq4A}$$
If $n \gt 1$, substituting $x_n = x_{n-1}^3 - 2x_{n-1}^2 + 2$ gives
$$\begin{equation}\begin{aligned}
(x_{n-1}^3 - 2x_{n-1}^2)^3 & \equiv (x_{n-1}^2(x_{n-1} - 2))^3 \\
& \equiv x_{n-1}^6(x_{n-1} - 2)^3 \\
& \equiv 1 \pmod{p}
\end{aligned}\end{equation}\tag{5}\label{eq5A}$$
Repeating the substitutions for each $x_i - 2 = x_{i-1}^2(x_{i-1} - 2)$ for $i$ from $n - 1$ down to $2$ gives a product of $0$ or more $x_{i-1}^6$ values multiplied by $(x_1 - 2)^3 = 3^3$, i.e.,
$$\left(\prod_{i=1}^{n-1}x_i\right)^6(3^3) \equiv 1 \pmod{p} \tag{6}\label{eq6A}$$
Multiplying both sides by $3$ gives
$$\left(\left(\prod_{i=1}^{n-1}x_i\right)^3 3^2\right)^2 \equiv 3 \pmod{p} \tag{7}\label{eq7A}$$
This shows $3$ is also a quadratic residue. Thus, $3^{-1}(-3) \equiv -1 \pmod{p}$ is a quadratic residue, so there's an integer $x$ with $x^2 \equiv -1 \pmod{p} \implies x^4 \equiv 1 \pmod{p}$. Therefore, $4$ is the multiplicative order of $x$ modulo $p$, so $4 \mid p -1 \implies p \equiv 1 \pmod{4}$. This proves that $p$ cannot be of the form $4k + 3$.
