AbstractIn this paper we obtain new upper bound estimates for the number of solutions of the congruence
$$\begin{equation}
x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in \mathcal{U},
\end{equation}$$
for certain ranges of H and |${\mathcal U}$|, where ${\mathcal U}$ is a subset of the field of residue classes modulo p having small multiplicative doubling. We then use these estimates to show that the number of solutions of the congruence
$$\begin{equation}
x^n\equiv \lambda\pmod p; \quad x\in \mathbb{N}, \quad L<x<L+p/n,
\end{equation}$$
is at most $p^{\frac{1}{3}-c}$ uniformly over positive integers n, λ and L, for some absolute constant c > 0. This implies, in particular, that if f(x) ∈ $\mathbb{Z}$[x] is a fixed polynomial without multiple roots in $\mathbb{C}$, then the congruence xf(x) ≡ 1 (mod p), x ∈ $\mathbb{N}$, x ⩽ p, has at most $p^{\frac{1}{3}-c}$ solutions as p → ∞, improving some recent results of Kurlberg, Luca and Shparlinski and of Balog, Broughan and Shparlinski. We use our results to show that almost all the residue classes modulo p can be represented in the form xgy (mod p) with positive integers x < p5/8+ϵ and y < p3/8. Here g denotes a primitive root modulo p. We also prove that almost all the residue classes modulo p can be represented in the form xyzgt (mod p) with positive integers x, y, z, t < p1/4+ϵ.
On the least square-free primitive root modulo p
