Waring type congruences involving factorials modulo a prime (Q869258)

The authors show that for a sufficiently large prime number \(p\) every residue class \(\lambda\) modulo \(p\) can be represented as \(\sum_{i=1}^5 m_i!n_i!\equiv\lambda\pmod p\), where \(m_i, n_i\ll p^{27/28}\), improving on a result by \textit{M. Z. Garaev, F. Luca} and \textit{I. E. Shparlinski} [J. Reine Angew. Math. 584, 29--44 (2004; Zbl 1071.11051)]. The improvement stems from the surprising observation that a certain mean value can be estimated with high precision, if we impose the restrictions \(m_1=m_2=m_3-1=m_4-1\). The authors also show that every residue class \(\lambda\) can be written as the sum of \(\ll\log^3 p\) factorials; however, this falls still short of the bound \((\log p/\log\log p)^2\), which follows from the fact that every integer \(<n!\) can be written as the sum of \(\leq\frac{n(n-1)}{2}\) factorials.