Additive congruences with factorials modulo a prime

Abstract

Let $p$ be a large prime number. We prove that any integer $\lambda$ modulo $p$ can be represented in the form $$ m!n! +\sum_{i=1}^{47}n_i!\equiv \lambda \pmod p, $$ 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 $\lambda$ can be represented in the form $$ m_1!n_1! +m_2!n_2!+m_3!n_3! +m_4!n_4!+m_5!n_5! \equiv \lambda \pmod p $$ 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-Pl\"unnecke inequality from additive combinatorics.