# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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## Character sums and congruences with $n!$HTML articles powered by AMS MathViewer

by Moubariz Z. Garaev, Florian Luca and Igor E. Shparlinski
Trans. Amer. Math. Soc. 356 (2004), 5089-5102 Request permission

## 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