# On the number of solutions of $pq^m \equiv k\bmod{n}$

It has been shown here that there exist positive integers $$n,k$$ such that for all primes $$p$$ and integers $$q\ge 1$$, $$m\ge 2$$, we have $$pq^m \not\equiv k\bmod{n}.$$

$$\text{ }$$

Hence, let $$f(n)$$ be the number of $$k \in \{1,2,\ldots,n\}$$ such that $$pq^m \equiv k\bmod{n}$$, for some prime $$p$$ and integers $$q\ge 1$$, $$m\ge 2$$. (In particular, $$f(n) \le n-1$$ for some $$n$$.)

Question. Is it true that there exists $$c>0$$ such that $$f(n) \ge cn$$ for all $$n$$?

Ps. Thanks to Erick Wong for the correction