Exercițiul 111

$x$, $y$ - prime, $x \gt y \textcolor{red}{\Rightarrow} x \gt 2 \textcolor{red}{\Rightarrow} x-impar \space \textcolor{red}{(1)}$
$p$ - prim, $x-impar \textcolor{red}{\Rightarrow} p \gt 2 \textcolor{red}{\Rightarrow} p-impar \space \textcolor{red}{(2)}$
Din (1) și (2) $\textcolor{red}{\Rightarrow} y-par \textcolor{red}{\Rightarrow} \boxed{y=2}$

$p \lt 1000 \textcolor{red}{\Rightarrow} x^{2n} \lt 1000 \textcolor{red}{\Rightarrow} x^n \lt 32$
Cum $x - prim, n \not= 0 \textcolor{red}{\Rightarrow} n \in \{1,2,3\}$

$\bold{Caz \space 1}$: $\boxed{n=1} \textcolor{red}{\Rightarrow} p=x^2+2$
$\hspace*{2em} \textcolor{red}{\Rightarrow} x \in \{3,5,7,11,13,17,19,23,29,31\}$
$\hspace*{2em} \textcolor{red}{\Rightarrow} p \in \{11,27,51,123,171,291,363,531,843,963\}$
$\hspace*{2em} \textcolor{red}{\Rightarrow} \boxed{p=11}$ (restul nu sunt prime, sunt $M_3$)

$\bold{Caz \space 2}$: $\boxed{n=2} \textcolor{red}{\Rightarrow} p=(x^2)^2+2$
$\hspace*{2em} \textcolor{red}{\Rightarrow} x \in \{3,5\}$
$\hspace*{2em} \textcolor{red}{\Rightarrow} p \in \{83, 627\}$
$\hspace*{2em} \textcolor{red}{\Rightarrow} \boxed{p=83}$ ($627$ nu este prim, este $M_3$)

$\bold{Caz \space 3}$: $n=3 \textcolor{red}{\Rightarrow} p=(x^3)^2+2$
$\hspace*{2em} \textcolor{red}{\Rightarrow} x=3$
$\hspace*{2em} \textcolor{red}{\Rightarrow} p=(3^3)^2+2 = 731=17 \cdot 43 \space (\not = prim)$

În total două numere speciale: $\boxed{p \in \{11, 83\}}.$