Waring type congruences involving factorials modulo a prime

摘要

In this paper we prove that any residue class λ modulo a large prime number p can be represented in the form $$ {sumlimits_{i = 1}^5 {m_{i} !n_{i} ! equiv lambda quad (bmod ;p)} } $$ for some positive integers m 1, n 1,... ,m 5, n 5 of the size O(p 27/28). This improves one of the results from [6] on representability of λ modulo p in the form $$ {sumlimits_{i = 1}^7 {m_{i} !n_{i} ! equiv lambda quad } }(bmod ;p) $$ with $maxnolimits _{{1 leq i leq 7}} { m_{i} ,n_{i} } , = ,O(p^{{{{33/34}} }} )$ . We also prove that any residue class modulo p can be represented in the form $n_{1} ! + cdots + n_{{ell }} !quad (bmod ;p)$ with ${ell }, = ,O(log ^{3} p)$ . This improves the result of [7].