Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T04:59:12.505Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE CONSECUTIVE POWERS OF A PRIMITIVE ROOT: GAPS AND EXPONENTIAL SUMS

Part of: Sequences and sets Finite fields and commutative rings (number-theoretic aspects) Elementary number theory

Published online by Cambridge University Press:  08 November 2011

Sergei V. Konyagin  and
Igor E. Shparlinski
Sergei V. Konyagin
Affiliation:
Steklov Mathematical Institute, 8 Gubkin Street, Moscow, 119991, Russia (email: [email protected])
Igor E. Shparlinski
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia (email: [email protected])
Get access

Abstract

For a primitive root g modulo a prime p≥1 we obtain upper bounds on the gaps between the residues modulo p of the N consecutive powers agn, n=1,…,N, which is uniform over all integers a with gcd (a,p)=1.

MSC classification

Secondary: 11A07: Congruences; primitive roots; residue systems 11B05: Density, gaps, topology 11B50: Sequences (mod $m$) 11T23: Exponential sums
Type
Research Article
Information
Mathematika , Volume 58 , Issue 1 , January 2012 , pp. 11 - 20
Copyright
Copyright © University College London 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bourgain, J., Ford, K., Konyagin, S. V. and Shparlinski, I. E., On the divisibility of Fermat quotients. Michigan Math. J. 59 (2010), 313–328.CrossRefGoogle Scholar
[2]Bourgain, J. and Garaev, M. Z., On a variant of sum–product estimates and explicit exponential sum bounds in prime fields. Math. Proc. Cambridge Philos. Soc. 146 (2008), 1–21.CrossRefGoogle Scholar
[3]Bourgain, J., Konyagin, S., Pomerance, C. and Shparlinski, I. E., On the smallest pseudopower. Acta Arith. 140 (2009), 43–55.CrossRefGoogle Scholar
[4]Bourgain, J., Konyagin, S. V. and Shparlinski, I. E., Product sets of rationals, multiplicative translates of subgroups in residue rings and fixed points of the discrete logarithm. Int. Math. Res. Not. IMRN 2008 (2008), 1–29, Article ID rnn090.Google Scholar
[5]Bourgain, J., Konyagin, S. V. and Shparlinski, I. E., Corrigenda to: Product sets of rationals, multiplicative translates of subgroups in residue rings and fixed points of the discrete logarithm. Int. Math. Res. Not. IMRN 2009 (2009), 3146–3147.CrossRefGoogle Scholar
[6]Cobeli, C., Gonek, S. and Zaharescu, A., On the distribution of small powers of a primitive root. J. Number Theory 88 (2001), 49–58.CrossRefGoogle Scholar
[7]Iwaniec, H. and Kowalski, E., Analytic Number Theory, American Mathematical Society (Providence, RI, 2004).Google Scholar
[8]Konyagin, S. V. and Shparlinski, I. E., Character Sums with Exponential Functions and Their Applications, Cambridge University Press (Cambridge, 1999).CrossRefGoogle Scholar
[9]Korobov, N. M., On the distribution of digits in periodic fractions. Mat. Sb. 89 (1972), 654–670 (in Russian).Google Scholar
[10]Montgomery, H. L., Distribution of small powers of a primitive root. In Advances in Number Theory, Clarendon Press (Oxford, 1993), 137–149.CrossRefGoogle Scholar
[11]Rudnick, Z. and Zaharescu, A., The distribution of spacings between small powers of a primitive root. Israel J. Math. 120 (2000), 271–287.CrossRefGoogle Scholar
[12]Shparlinski, I. E., On the value set of Fermat quotients. Proc. Amer. Math. Soc. (to appear).Google Scholar
[13]Vâjâitu, M. and Zaharescu, A., Differences between powers of a primitive root. Int. J. Math. Math. Sci. 29 (2002), 325–331.CrossRefGoogle Scholar