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On runs of consecutive quadratic residues and quadratic nonresidues

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Abstract

Based on results of Weil and of Burgess, we have obtained a boundK(l) such that all primesp ≧K(l) have a sequence of at leastl consecutive quadratic residues and a sequence of at leastl consecutive nonresidues in the interval [1,p − 1]. The bound forl=9 being 414463, we have computed, for primes less than 420000, the lengths of the longest sequences of consecutive residues and of nonresidues. We present these data and make some observations concerning them. One of the observations is that there is an observed difference in the length of the maximal sequence between primes congruent to 1 (mod 4) and primes congruent to 3 (mod 4).

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References

  1. Duncan A. Buell and Richard H. Hudson,Computation of runs of consecutive quadratic residues and quadratic nonresidues, Louisiana State University Computer Science Technical Report # 83-017.

  2. D. A. Burgess,The distribution of quadratic residues and non-residues, Mathematika, 4 (1957), 106–112.

    Google Scholar 

  3. D. A. Burgess,On character sums and primitive roots, Proc. London Math. Soc., (3) 12 (1962), 179–192.

    Google Scholar 

  4. Richard H. Hudson,A bound for the first occurrence of three consecutive integers with equal quadratic character, Duke J. of Math., 40 (1973), 33–39.

    Google Scholar 

  5. Richard H. Hudson,On the first occurrence of certain patterns of quadratic residues and non-residues, Israel J. of Math., 44 (1983), 23–32.

    Google Scholar 

  6. Ernst Jacobsthal,Anwendungen einer Formel aus der Theorie der quadratischen Reste, Diss., Univ. Berlin, 1906, 1–39.

    Google Scholar 

  7. D. H. Lehmer and Emma Lehmer,On runs of residues, Proc. Amer. Math. Soc., 13 (1962), 102–106.

    Google Scholar 

  8. Emma Lehmer,Patterns of power residues, J. Number Theory, to appear.

  9. W. H. Mills,Bounded consecutive residues and related problems, Proc. Symp. Pure Math., American Mathematical Society, 8 (1965), 170–174.

    Google Scholar 

  10. Ronald E. Walpole,Introduction to Statistics, Macmillan, New York, 1968.

    Google Scholar 

  11. A. Weil,Sur les courbes algébriques et les varietés qui s'en deduisent, Actualités Math. Sci., No. 1041 (Paris, 1945), deuxième partie, section IV.

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Authors and Affiliations

  1. Computer Science Department, Louisiana State University, 70803, Baton Rouge, Louisiana, U.S.A.

    Duncan A. Buell & Richard H. Hudson

  2. Mathematics and Statistics Department, University of South Carolina, 29208, Columbia, South Carolina, U.S.A.

    Duncan A. Buell & Richard H. Hudson

Authors
  1. Duncan A. Buell
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  2. Richard H. Hudson
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Buell, D.A., Hudson, R.H. On runs of consecutive quadratic residues and quadratic nonresidues. BIT 24, 243–247 (1984). https://doi.org/10.1007/BF01937490

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  • DOI: https://doi.org/10.1007/BF01937490

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