In this paper we obtain new upper bound estimates for the number of solutions of the congruence equation x≡ y r p; x,y∈ N, x,y≤ H, r∈ U, equation for certain ranges of H and | U |, where 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 equation xⁿ≡ λ p; x∈ N, L<x<L+p/n, equation is at most p1/3-c uniformly over positive integers n , λ and L , for some absolute constant c > 0. This implies, in particular, that if f ( x ) ∈ Z [ x ] is a fixed polynomial without multiple roots in C , then the congruence x f ( x ) ≡ 1 (mod p ), x ∈ N , x ⩽ p , has at most p1/3-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 xg y (mod p ) with positive integers x < p 5/8+ϵ and y < p 3/8 . 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 xyzg t (mod p ) with positive integers x, y, z, t < p 1/4+ϵ .
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Cilleruelo et al. (2016) studied this question.