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Waring type congruences involving factorials modulo a prime

Authors:

M. Z. Garaev V. C. Garcia

Affiliation:

(1) Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacán, México

Abstract:

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 m₁, n₁,... ,m₅, n₅ 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]. Received: 27 March 2006

Keywords:

Mathematics Subject Classification (2000). 11A07 11B65

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