Character sums and congruences with n!

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

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

23 Citations (Scopus)


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 ≪ p1/2+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 ≡ nI!...n7! (mod p), where max{ni

Original languageEnglish
Pages (from-to)5089-5102
Number of pages14
JournalTransactions of the American Mathematical Society
Issue number12
Publication statusPublished - Dec 2004


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