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Waring type congruences involving factorials modulo a prime

Authors:

M Z Garaev V C Garcia

Institution:

(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

${\sum\limits_{i = 1}^5 {m_{i} !n_{i} ! \equiv \lambda \quad (\bmod p)} }$

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

${\sum\limits_{i = 1}^7 {m_{i} !n_{i} ! \equiv \lambda \quad } }(\bmod p)$

with $\max\nolimits _{{1 \leq i \leq 7}} \{ m_{i} ,n_{i} \} \, = \,O(p^{{{{33/34}} }} )$ . 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)" target="_blank">Mathematics Subject Classification (2000) 11A07 11B65

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