On multiplicative congruences

Abstract

Let $\epsilon$ be a fixed positive quantity, $m$ be a large integer, $x_j$ denote integer variables. We prove that for any positive integers $N_1,N_2,N_3$ with $N_1N_2N_3>m^{1+\epsilon},$ the set $$ \{x_1x_2x_3 \pmod m: \quad x_j\in [1,N_j] \} $$ contains almost all the residue classes modulo $m$ (i.e., its cardinality is equal to $m+o(m)$). We further show that if $m$ is cubefree, then for any positive integers $N_1,N_2,N_3,N_4$ with $N_1N_2N_3N_4>m^{1+\epsilon},$ the set $$ \{x_1x_2x_3x_4 \pmod m: \quad x_j\in [1,N_j] \} $$ also contains almost all the residue classes modulo $m.$ Let $p$ be a large prime parameter and let $p>N>p^{63/76+\epsilon}.$ We prove that for any nonzero integer constant $k$ and any integer $\lambda\not\equiv 0\pmod p$ the congruence $$ p_1p_2(p_3+k)\equiv \lambda\pmod p $$ admits $(1+o(1))\pi(N)^3/p$ solutions in prime numbers $p_1, p_2, p_3\le N.$