On multiplicative congruences
| dc.creator | Garaev, M. Z. | |
| dc.date | 2008-07-27 | |
| dc.date | 2008-08-11 | |
| dc.date.accessioned | 2026-07-07T09:55:37Z | |
| dc.date.available | 2026-07-07T09:55:37Z | |
| dc.description | Let $ε$ 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 $N_1N_2N_3>m^{1+ε},$ 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 $N_1N_2N_3N_4>m^{1+ε},$ 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 $p>N>p^{63/76+ε}.$ We prove that for any nonzero integer constant $k$ and any integer $λ\not\equiv 0\pmod p$ the congruence $$ p_1p_2(p_3+k)\equiv λ\pmod p $$ admits $(1+o(1))π(N)^3/p$ solutions in prime numbers $p_1, p_2, p_3\le N.$ | |
| dc.description | Minor typographical corrections | |
| dc.identifier | https://arxiv.org/abs/0807.4318 | |
| dc.identifier | http://arxiv.org/abs/0807.4318 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/166688 | |
| dc.subject | Number Theory | |
| dc.subject | 11L40 | |
| dc.title | On multiplicative congruences | |
| dc.type | text |