M. Garaev

activity

20042025

most citedCongruences involving product of intervals and sets with small multiplicative doubling modulo a prime and applications

30 citations · 125 across the 33 of their papers we have counts for

collaborators

36 papers

math.NT2025

Additive congruences with factorials modulo a prime

Moubariz Z. Garaev, Julio C. Pardo

Let $p$ be a large prime number. We prove that any integer $λ$ modulo $p$ can be represented in the form $m! n! + i = 1 \sum 47 n_{i}! \equiv λ (mod p),$ with $\max\{m,n,n_1,\ldots,…

math.NT2024

Binary and ternary congruences involving intervals and sets modulo a prime

Moubariz Z. Garaev, Julio C. Pardo, Igor E. Shparlinski

Let $s$ be a fixed positive integer constant, $ε$ be a fixed small positive number. Then, provided that a prime $p$ is large enough, we prove that for any set $\{{\mathca…

math.NT2023

On a family of sparse exponential sums

Moubariz Garaev, Zeev Rudnick, Igor Shparlinski

We investigate exponential sums modulo primes whose phase function is a sparse polynomial, with exponents growing with the prime. In particular, such sums model those which appear…

math.NT2023

On some congruences and exponential sums

Moubariz Z. Garaev, Igor E. Shparlinski

Let $ε > 0$ be a fixed small constant, $F_{p}$ be the finite field of $p$ elements for prime $p$ . We consider additive and multiplicative problems in ${\mathbb F}_p…

math.NT2023

On the distribution of modular inverses from short intervals

Moubariz Z. Garaev, Igor E. Shparlinski

For a prime number $p$ and integer $x$ with $g cd (x, p) = 1$ let $x$ denote the multiplicative inverse of $x$ modulo $p .$ In the present paper we are interested in the prob…

math.NT2022

On integer values of sum and product of three positive rational numbers

M. Z. Garaev

In 1997 we proved that if $n$ is of the form $4 k, 8 k - 1 or 2^{2 m + 1} (2 k - 1) + 3,$ where $k, m \in N,$ then there are no positive rational numbers $x,y,z…