Character sums and congruences with n!

Moubariz Z. Garaev; Florian Luca; Igor E. Shparlinski

doi:10.1090/S0002-9947-04-03612-8

Character sums and congruences with n!

Moubariz Z. Garaev^*
, Florian Luca
, Igor E. Shparlinski

^*Corresponding author for this work

School of Computing

Research output: Contribution to journal › Article › peer-review

25 Citations (Scopus)

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 ≪ p^1/2+ε such that n! is a primitive root modulo p. We also show that every nonzero congruence class a ≢ 0 (mod p) can be represented as a product of 7 factorials, a ≡ n_I!...n₇! (mod p), where max{n_i

Original language	English
Pages (from-to)	5089-5102
Number of pages	14
Journal	Transactions of the American Mathematical Society
Volume	356
Issue number	12
DOIs	https://doi.org/10.1090/S0002-9947-04-03612-8
Publication status	Published - Dec 2004

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Character sums and congruences with n!

Abstract

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