   ISSN 1088-6850(online) ISSN 0002-9947(print)

Character sums and congruences with $n!$

Authors: Moubariz Z. Garaev, Florian Luca and Igor E. Shparlinski
Journal: Trans. Amer. Math. Soc. 356 (2004), 5089-5102
MSC (2000): Primary 11A07, 11B65, 11L40
DOI: https://doi.org/10.1090/S0002-9947-04-03612-8
Published electronically: June 29, 2004
MathSciNet review: 2084412
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Abstract: We estimate character sums with $n!$, on average, and individually. These bounds are used to derive new results about various congruences modulo a prime $p$ and obtain new information about the spacings between quadratic nonresidues modulo $p$. In particular, we show that there exists a positive integer $n\ll p^{1/2+\varepsilon }$ such that $n!$ is a primitive root modulo $p$. We also show that every nonzero congruence class $a \not \equiv 0 \pmod p$ can be represented as a product of 7 factorials, $a \equiv n_1! \ldots n_7! \pmod p$, where $\max \{n_i \ |\ i=1,\ldots , 7\}=O(p^{11/12+\varepsilon })$, and we find the asymptotic formula for the number of such representations. Finally, we show that products of 4 factorials $n_1!n_2!n_3!n_4!,$ with $\max \{n_1, n_2, n_3, n_4\}=O(p^{6/7+\varepsilon })$ represent “almost all” residue classes modulo p, and that products of 3 factorials $n_1!n_2!n_3!$ with $\max \{n_1, n_2, n_3\}=O(p^{5/6+\varepsilon })$ are uniformly distributed modulo $p$.

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Moubariz Z. Garaev
Affiliation: Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58180, Morelia, Michoacán, México
MR Author ID: 632163
Email: garaev@matmor.unam.mx

Florian Luca
Affiliation: Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58180, Morelia, Michoacán, México
MR Author ID: 630217
Email: fluca@matmor.unam.mx

Igor E. Shparlinski
Affiliation: Department of Computing, Macquarie University, Sydney, New South Wales 2109, Australia
MR Author ID: 192194
Email: igor@ics.mq.edu.au

Received by editor(s): September 29, 2003
Published electronically: June 29, 2004