I encountered the following while solving another problem and need some help.
Let $p,d$ be relatively prime natural numbers. I think that for many of $(p,d)$ tuples, numbers in set $\mathbb{N}_d = \{0,1,2,\cdots,d-1\}$ which are residue of one of the $1, p, p^2, \cdots$ modulo $d$ are kind of widely, uniformly distributed. More precisely, let's call $n\in\mathbb{N}_d$ which there exists natural number $u$ such that $n \equiv p^u\;(\text{mod} \;d)$, a good number. Are good numbers 'widely' distributed?
What I actually need for the original problem is that, there are only few $(p,d,c)$ such that for the smallest non trivial divisor of $p$, let's say $t$, some numbers called '$c$-good numbers', which are $n\in\mathbb{N}_d$ such that there exists $u$ with $n\equiv cp^u\;(\text{mod }d)$, are all smaller than the rational number $d/t$.
This was what I mean by 'widely distributed'. Are there some known theorems or results that imply or mean the wide distribution of powers in the residue system, which may potentially be useful for the problem?