Additive congruences with factorials modulo a prime
Abstract
Let p p be a large prime number. We prove that any integer λ \lambda modulo p p can be represented in the form m ! n ! + ∑ i = 1 47 n i ! ≡ λ ( mod p ) , \begin{equation*} m!n! +\sum _{i=1}^{47}n_i!\equiv \lambda \pmod p, \end{equation*} with max { m , n , n 1 , … , n 47 } ≪ p 1300 / 1301 \max \{m,n,n_1,\ldots ,n_{47}\}\ll p^{1300/1301} . This improves the exponent 1350 / 1351 1350/1351 of Garaev, Luca and Shparlinski (2005). Furthermore, we prove that any integer λ \lambda can be represented in the form m 1 ! n 1 ! + m 2 ! n 2 ! + m 3 ! n 3 ! +
Fields of Study & Subjects
Analytic Number Theory Research
Advanced Mathematical Identities
Algebraic Geometry and Number Theory
Publication details
- Journal / source
- Proceedings of the American Mathematical Society
- Volume
- 154
- Issue
- 10
- Pages
- 4179–4190
- Publication date
- 17 June 2026
- Language
- EN