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Character sums and congruences with

Authors:	Moubariz Z Garaev Florian Luca Igor E Shparlinski

Institution:	Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58180, Morelia, Michoacán, México ; Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58180, Morelia, Michoacán, México ; Department of Computing, Macquarie University, Sydney, New South Wales 2109, Australia

Abstract:	We estimate character sums with , on average, and individually. These bounds are used to derive new results about various congruences modulo a prime and obtain new information about the spacings between quadratic nonresidues modulo . In particular, we show that there exists a positive integer $n\ll p^{1/2+\varepsilon}$ such that is a primitive root modulo . 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 $\max \{n_i \vert i=1,\ldots, 7\}=O(p^{11/12+\varepsilon})$ , and we find the asymptotic formula for the number of such representations. Finally, we show that products of 4 factorials 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 with $\max\{n_1, n_2, n_3\}=O(p^{5/6+\varepsilon})$ are uniformly distributed modulo .

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