On multiplicative congruences

TL;DR

This paper investigates the distribution of products modulo m, showing that large enough sets of integer products cover almost all residue classes, and extends results to prime solutions of certain multiplicative congruences.

Contribution

It proves that product sets with sufficiently large ranges cover almost all residue classes modulo m, and establishes the existence of prime solutions to specific multiplicative congruences.

Findings

Product sets with N_1N_2N_3 > m^{1+ε} cover almost all residue classes modulo m

For cubefree m, fourfold product sets also cover almost all residue classes

Prime solutions to p_1p_2(p_3+k) ≡ λ mod p exist with high density for large primes

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 $p_1, p_2, p_3\le N.$