TY - JOUR

T1 - Character sums and congruences with n!

AU - Garaev, Moubariz Z.

AU - Luca, Florian

AU - Shparlinski, Igor E.

PY - 2004/12

Y1 - 2004/12

N2 - 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

AB - 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

UR - http://www.scopus.com/inward/record.url?scp=10044289130&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-04-03612-8

DO - 10.1090/S0002-9947-04-03612-8

M3 - Article

AN - SCOPUS:10044289130

VL - 356

SP - 5089

EP - 5102

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 12

ER -