2 problems

For a prime $p$, consider the two-cycle equations … the self-power equation $x^x\equiv c\pmod p$ for fixed $c$, and the collision equation … Here $T(p)$ counts solutions to the two…

Let $p$ be prime and $e\geq 1$. Consider solutions $(g,x)$ of … with $g,x\in\{1,\dots,p^e\}$ and $p\nmid g,x$. Fixed-point counting conjecture. The number of these solutions is asy…