Type: Article

On multiplicative congruences

Garaev, M. Z.;

Abstract:

Let 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-1, N-2, N-3 > m(1+epsilon), the set {x(1)x(2)x(3) (mod m) : x(j) epsilon [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-1, N-2, N-3, N-4 > m(1+epsilon), the set {x(1)x(2)x(3)x(4) (mod m) : x(j) epsilon [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 the congruence lambda not equivalent to 0 (mod p) the congruence p(1)p(2)(p(3)+k) lambda (mod p) admits (1 + o(1))pi(N)(3)/p solutions in prime numbers p(1), p(2), p(3) <= N.