Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the value set of Fermat quotients


Author: Igor E. Shparlinski
Journal: Proc. Amer. Math. Soc. 140 (2012), 1199-1206
MSC (2000): Primary 11A07, 11L07
DOI: https://doi.org/10.1090/S0002-9939-2011-11203-6
Published electronically: August 2, 2011
MathSciNet review: 2869105
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We obtain an upper bound $ p^{463/252+o(1)}$ on the smallest $ L$ such that the set of the first $ L$ Fermat quotients modulo a prime $ p$ represents all residues modulo $ p$.


References [Enhancements On Off] (What's this?)

  • 1. J. Bourgain, K. Ford, S. V. Konyagin and I. E. Shparlinski, `On the divisibility of Fermat quotients', Michigan Math. J., 59 (2010), 313-328. MR 2677624
  • 2. J. Bourgain, S. V. Konyagin and I. E. Shparlinski, `Product sets of rationals, multiplicative translates of subgroups in residue rings and fixed points of the discrete logarithm', Intern. Math. Research Notices, 2008 (2008), Article ID rnn090, 1-29. MR 2439546 (2009i:11007)
  • 3. J. Bourgain, S. V. Konyagin and I. E. Shparlinski, `Corrigenda to: Product sets of rationals, multiplicative translates of subgroups in residue rings and fixed points of the discrete logarithm', Intern. Math. Research Notices, 2009 (2009), 3146-3147. MR 2533800 (2010i:11006)
  • 4. J. Bourgain, S. Konyagin, C. Pomerance and I. E. Shparlinski, `On the smallest pseudopower', Acta Arith., 140 (2009), 43-55. MR 2557852 (2010j:11127)
  • 5. Z. Chen, A. Ostafe and A. Winterhof, `Structure of pseudorandom numbers derived from Fermat quotients', Proc. Intern. Workshop on the Arith. of Finite Fields, Istanbul, 2010. Lect. Notes in Comp. Sci., vol. 6087, Springer-Verlag, Berlin, 2010, 73-85.
  • 6. R. Ernvall and T. Metsänkylä, `On the $ p$-divisibility of Fermat quotients', Math. Comp., 66 (1997), 1353-1365. MR 1408373 (97i:11003)
  • 7. W. L. Fouché, `On the Kummer-Mirimanoff congruences', Quart. J. Math. Oxford, 37 (1986), 257-261. MR 854625 (88a:11022)
  • 8. D. Gomez and A. Winterhof, `Multiplicative character sums of Fermat quotients and pseudorandom sequences', Period. Math. Hungarica (to appear).
  • 9. A. Granville, `Some conjectures related to Fermat's Last Theorem', Number Theory, de Gruyter, New York, 1990, 177-192. MR 1106660 (92k:11036)
  • 10. A. Granville, `On pairs of coprime integers with no large prime factors', Expos. Math., 9 (1991), 335-350. MR 1137813 (92m:11095)
  • 11. D. R. Heath-Brown, `An estimate for Heilbronn's exponential sum', Analytic Number Theory: Proc. Conf. in Honor of Heini Halberstam, Birkhäuser, Boston, 1996, 451-463. MR 1409372 (97k:11120)
  • 12. D. R. Heath-Brown and S. V. Konyagin, `New bounds for Gauss sums derived from $ k$th powers, and for Heilbronn's exponential sum', Quart. J. Math., 51 (2000), 221-235. MR 1765792 (2001h:11106)
  • 13. Y. Ihara, `On the Euler-Kronecker constants of global fields and primes with small norms', Algebraic Geometry and Number Theory, Progress in Math., Vol. 253, Birkhäuser Boston, Boston, MA, 2006, 407-451. MR 2263195 (2007h:11127)
  • 14. S. V. Konyagin and I. E. Shparlinski, Character sums with exponential functions and their applications, Cambridge Univ. Press, Cambridge, UK, 1999. MR 1725241 (2000h:11089)
  • 15. H. W. Lenstra, `Miller's primality test', Inform. Process. Lett., 8 (1979), 86-88. MR 520273 (80c:10008)
  • 16. A. Ostafe and I. E. Shparlinski, `Pseudorandomness and dynamics of Fermat quotients', SIAM J. Discr. Math., 25 (2011), 50-71.
  • 17. I. E. Shparlinski, `Character sums with Fermat quotients', Quart. J. Math. (to appear).
  • 18. I. E. Shparlinski, `Bounds of multiplicative character sums with Fermat quotients of primes', Bull. Aust. Math. Soc. 83 (2011), 456-462.
  • 19. H. S. Vandiver, `An aspect of the linear congruence with applications to the theory of Fermat's quotient', Bull. Amer. Math. Soc., 22 (1915), 61-67. MR 1559712

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11A07, 11L07

Retrieve articles in all journals with MSC (2000): 11A07, 11L07


Additional Information

Igor E. Shparlinski
Affiliation: Department of Computing, Macquarie University, Sydney, NSW 2109, Australia
Email: igor.shparlinski@mq.edu.au

DOI: https://doi.org/10.1090/S0002-9939-2011-11203-6
Keywords: Fermat quotients, value set, preimage
Received by editor(s): December 20, 2010
Received by editor(s) in revised form: January 2, 2011
Published electronically: August 2, 2011
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2011 American Mathematical Society

American Mathematical Society