Julio C. Pardo | Searcharxiv

arXiv2025

Additive congruences with factorials modulo a prime

Let $p$ be a large prime number. We prove that any integer $λ$ modulo $p$ can be represented in the form $$ m!n! +\sum_{i=1}^{47}n_i!\equiv λ\pmod p, $$ with $\max\{m,n,n_1,\ldots,n_{47}\}\ll p^{1300/1301}.$ This improves the exponent $1350/1351$ of Garaev, Luca and Shparlinski (2005). Furthermore, we prove that any integer $λ$ can be represented in the form $$ m_1!n_1! +m_2!n_2!+m_3!n_3! +m_4!n_4!+m_5!n_5! \equiv λ\pmod p $$ with $\max\{m_1,n_1,\ldots,m_5,n_5\}\le p^{97/113 +o(1)}.$ This improves the exponent $27/28$ of Garaev and Garcia (2007). The proofs of these two results are based on the recent work of Grebennikov, Sagdeev, Semchankau and Vasilevskii (2024). We also obtain some lower bound estimates on the cardinality of the product set of two factorials modulo a prime. For instance, we prove that if $N<p^{3/5},$ then $$ \#\{m!n!\pmod p; \, 1\le m,n\le N\}\gg N^{1-o(1)}. $$ The proof of this result is based on works of Banks and Shparlinski (2020), Cilleruelo and Garaev (2016), and the work of Katz and Shen (2008) related to the Ruzsa-Plünnecke inequality from additive combinatorics.

math.NT↗

arXiv2024

Binary and ternary congruences involving intervals and sets modulo a prime

Let $s$ be a fixed positive integer constant, $\varepsilon$ be a fixed small positive number. Then, provided that a prime $p$ is large enough, we prove that for any set $\{{\mathcal M}\subseteq \mathbb F_p^*$ of size $|{\mathcal M}|= \lfloor p^{14/29}\rfloor$ and integer $H=\lfloor p^{14/29+\varepsilon}\rfloor$, any integer $λ$ can be represented in the form $$ \frac{m_1}{x_1^s}+\frac{m_2}{x_2^s}+\frac{m_3}{x_3^s}\equiv λ\bmod p, $$ with $$ m_i\in {\mathcal M}, \quad 1\le x_i\le H, \qquad i=1,2,3. $$ When $s=1$ we show that for almost all primes $p$ the following holds: if $|{\mathcal M}|= \lfloor p^{1/2}\rfloor$ and $H=\lfloor p^{1/2}(\log p)^{6+\varepsilon}\rfloor$, then any integer $λ$ can be represented in the form $$ \frac{m_1}{x_1}+\frac{m_2}{x_2}\equiv λ\bmod p, $$ with $$ m_i\in {\mathcal M}, \quad 1\le x_i\le H, \qquad i=1,2. $$

math.NT↗