On multiplicative congruences

Abstract

Let ${\varepsilon}$ 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 ${N_1N_2N_3 > m^{1+\varepsilon}, }$ the set

$$\{x_1x_2x_3 \quad ({\rm mod}\,m): \quad x_j\in [1,N_j]\}$$

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 ${ N_1N_2N_3N_4 > m^{1+\varepsilon}, }$ the set

$$\{x_1x_2x_3x_4 \quad ({\rm mod}\,m): \quad x_j\in [1,N_j]\}$$

also contains almost all the residue classes modulo m. Let p be a large prime parameter and let ${p > N > p^{63/76+\varepsilon}.}$ We prove that for any nonzero integer constant k and any integer ${\lambda\not\equiv 0 \,\, ({\rm mod}\,p)}$ the congruence

$$p_1p_2(p_3+k)\equiv \lambda \quad ({\rm mod}\, p) $$

admits (1 + o(1))π(N)³/p solutions in prime numbers p ₁, p ₂, p ₃ ≤ N.