Archiv der Mathematik

, Volume 95, Issue 6, pp 529–537 | Cite as

On pseudopoints of algebraic curves

  • Reza R. Farashahi
  • Igor E. Shparlinski


Following Kraitchik and Lehmer, we say that a positive integer n ≡ 1 (mod 8) is an x-pseudosquare if it is a quadratic residue for each odd prime px, yet it is not a square. We extend this definition to algebraic curves and say that n is an x-pseudopoint of a curve defined by f(U, V) = 0 (where \({f \in \mathbb{Z}[U, V]}\)) if for all sufficiently large primes px the congruence f(n, m) ≡ 0 (mod p) is satisfied for some m. We use the Bombieri bound of exponential sums along a curve to estimate the smallest x-pseudopoint, which shows the limitations of the modular approach to searching for points on curves.

Mathematics Subject Classification (2000)

11T23 14G05 


Algebraic curve Pseudopoint 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bach E. et al.: Results and estimates on pseudopowers. Math. Comp. 65, 1737–1747 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    D. J. Bernstein, Doubly focused enumeration of locally square polynomial values, High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, Fields Institute Communications 41, Amer. Math. Soc., 2004, 69–76.Google Scholar
  3. 3.
    Bombieri E.: On exponential sums in finite fields. Amer. J. Math. 88, 71–105 (1966)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bombieri E., Pila J.: The number of integral points on arcs and ovals. Duke Math. J. 59, 337–357 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bourgain J. et al.: On the smallest pseudopower. Acta Arith. 140, 43–55 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gao S., Rodrigues V.M.: Irreducibility of polynomials modulo p via Newton polytopes. J. Number Theory 101, 32–47 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hindry M., Silverman J.H.: Diophantine geometry. Springer-Verlag, Berlin (2000)zbMATHGoogle Scholar
  8. 8.
    H. Iwaniec and E. Kowalski, Analytic number theory, Colloquium Pubs. 53, Amer. Math. Soc., Providence, RI, 2004.Google Scholar
  9. 9.
    Konyagin S.V., Pomerance C., Shparlinski I.E.: On the distribution of pseudopowers. Canad. J. Math. 62, 582–594 (2010)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Lehmer D.H.: A sieve problem on “pseudo-squares”. Math. Tables and Other Aids to Computation 8, 241–242 (1954)MathSciNetGoogle Scholar
  11. 11.
    D. Lorenzini, An invitation to arithmetic geometry, Amer. Math. Soc., 1996.Google Scholar
  12. 12.
    A. Ostrowski, Zur arithmetischen Theorie der algebraischen Grössen, Nachr. K. Ges. Wiss. Göttingen (1919), 273–298.Google Scholar
  13. 13.
    Pila J.: Density of integer points on plane algebraic curves. Intern. Math. Research Notices 18, 903–912 (1996)CrossRefMathSciNetGoogle Scholar
  14. 14.
    C. Pomerance, and I. E. Shparlinski, On pseudosquares and pseudopowers, Combinatorial Number Theory, Proc. of Integers Conf. 2007, Walter de Gruyter, Berlin, 2009, 171–184.Google Scholar
  15. 15.
    Ruppert W.M.: Reducibility of polynomials f(x; y) modulo p. J. Number Theory 77, 62–70 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    A. Schinzel, On pseudosquares, New trends in probability and statistics, Palonga, 1996, 4, VSP, Utrecht, 1997, 213–220.Google Scholar
  17. 17.
    J. P. Sorenson, The pseudosquares prime sieve, Proc. 7th Algorithmic Number Theory Symp., Lect. Notes in Comput. Sci. 4076, Springer-Verlag, Berlin, 2006, 193–207.Google Scholar
  18. 18.
    Weil A.: On some exponential sums. Proc. Nat. Acad. Sci. USA 34, 204–207 (1948)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    K. Wooding and H. C. Williams, Doubly-focused enumeration of pseudosquares and pseudocubes, Proc. 7th Algorithmic Number Theory Symp., Lect. Notes in Comput. Sci. 4076, Springer-Verlag, Berlin, 2006, 208–211.Google Scholar
  20. 20.
    Zannier U.: On the reduction modulo p of an absolutely irreducible polynomial f(x; y). Arch. Math. 68, 129–138 (1997)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of Computing, Faculty of ScienceMacquarie UniversitySydneyAustralia

Personalised recommendations