On q-orders in primitive modular groups

Jacek Pomykała. "On q-orders in primitive modular groups." Acta Arithmetica 166.4 (2014): 397-404. <http://eudml.org/doc/279782>.

@article{JacekPomykała2014,
abstract = {We prove an upper bound for the number of primes p ≤ x in an arithmetic progression 1 (mod Q) that are exceptional in the sense that $ℤ*_p$ has no generator in the interval [1,B]. As a consequence we prove that if $Q > exp[c (log p)/(log B) (loglogp)]$ with a sufficiently large absolute constant c, then there exists a prime q dividing Q such that $ν_q(ord_p b) = ν_q(p-1)$ for some positive integer b ≤ B. Moreover we estimate the number of such q’s under suitable conditions.},
author = {Jacek Pomykała},
journal = {Acta Arithmetica},
keywords = {Dirichlet characters; -functions; least character nonresidues; Riemann Hypothesis; modular groups; orders; complex roots of unity; deterministic algorithms in cryptography},
language = {eng},
number = {4},
pages = {397-404},
title = {On q-orders in primitive modular groups},
url = {http://eudml.org/doc/279782},
volume = {166},
year = {2014},
}

TY - JOUR
AU - Jacek Pomykała
TI - On q-orders in primitive modular groups
JO - Acta Arithmetica
PY - 2014
VL - 166
IS - 4
SP - 397
EP - 404
AB - We prove an upper bound for the number of primes p ≤ x in an arithmetic progression 1 (mod Q) that are exceptional in the sense that $ℤ*_p$ has no generator in the interval [1,B]. As a consequence we prove that if $Q > exp[c (log p)/(log B) (loglogp)]$ with a sufficiently large absolute constant c, then there exists a prime q dividing Q such that $ν_q(ord_p b) = ν_q(p-1)$ for some positive integer b ≤ B. Moreover we estimate the number of such q’s under suitable conditions.
LA - eng
KW - Dirichlet characters; -functions; least character nonresidues; Riemann Hypothesis; modular groups; orders; complex roots of unity; deterministic algorithms in cryptography
UR - http://eudml.org/doc/279782
ER -