They're all PRIME!!

${ 23 }^{ 23 }\equiv { (-8) }^{ 23 }\quad (mod\quad 31)\\ \quad \quad \quad \equiv -{ (8) }^{ 23 }\quad (mod\quad 31)\\ \quad \quad \quad \equiv -{ { (2 }^{ 3 } })^{ 23 }\quad (mod\quad 31)\\ \quad \quad \quad \equiv -({ 2) }^{ 69 }\quad (mod\quad 31)\\ \quad \quad \quad \equiv -({ 2) }^{ 65 }.16\quad (mod\quad 31)\\ \quad \quad \quad \equiv -{ { (2 }^{ 5 }) }^{ 13 }.16\quad (mod\quad 31)\\ \quad \quad \quad \equiv -({ 32) }^{ 13 }.16\quad (mod\quad 31)\\ \quad \quad \quad \equiv -({ 1) }^{ 13 }.16\quad (mod\quad 31)\\ \quad \quad \quad \equiv -16\quad (mod\quad 31)\\ \quad \quad \quad \equiv 15\quad (mod\quad 31)$

88

00

$\begin{aligned} r &\equiv 23^{23} \pmod{31} \ &\blue{0 \leq r <31} \\ &\equiv (23^{22})(23) \pmod{31} \\ &\equiv (529^{11})(23) \pmod{31} \\ &\equiv (2^{11})(23) \pmod{31} \\ &\equiv (32^2)(2)(23) \pmod{31} \\ &\equiv (1)(2)(23) \pmod{31} \\ &\equiv 46 \pmod{31} \\ &\equiv \boxed{15} \pmod{31} \\ \end{aligned}$