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Results in Mathematics

, Volume 48, Issue 1–2, pp 109–123 | Cite as

On a conjecture of Ma

  • Florian Luca
  • Pantelimon Stănică
Article

Abstract

In this paper, we prove a result concerning a conjecture of Ma from diophantine equations, which is connected to an open problem on abelian difference sets of multiplier −1.

2000 Mathematics Subject Classification

05B10 11D45 11D72 

En]Keywords

Difference sets diophantine equations subspace theorem 

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References

  1. [1]
    R. Calderbank, ’On uniformly packed [n, n - k,4] codes over GF(q) and a class of caps in PG(k-1,q)’, J. London Math. Soc. 26 (1982), 365–384.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    R.D. Carmichael, ’On the numerical factors of arithmetic forms αn±βn’, Ann. of Math. 15 (1913), 30–70.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    J.-H. Evertse, ’The number of solutions of decomposable form equations’, Invent. Math. 122 (1995), 559–601.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    J.-H. Evertse, ’An improvement of the Quantitative Subspace Theorem’, Compos. Math. 101 (1996), 225–311.MathSciNetMATHGoogle Scholar
  5. [5]
    D. Jungnickel, ‘Difference Sets’, In J.Dinitz, D.R. Stinson (ed.) Contemporary Design Theory, A Colection of Surveys. Wiley-Interscience Series in Discrete Mathematics and Optimization (1992), 241-324.Google Scholar
  6. [6]
    M. Le and Q. Xiang, ’A result on Ma’s conjecture’, J. Combin. Theory Ser. A 73 (1996), 181–184.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    F. Luca, ’On the equation x2 = 4ym+n ± 4yn + 1’. Bull. Math. Soc. Sci Math. Roumanie 90 (1999), 231–235.Google Scholar
  8. [8]
    F. Luca, ’On the Diophantine equation x2 = 4qm − 4qn + 1’, Proc. Amer. Math. Soc. 131 (2003), 1339–1345.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    F. Luca, ’The Diophantine equation x2 = pa ± pb + 1’, Acta Arith. 112 (2004), 87–101.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    S.L. Ma, ’McFarland’s conjecture on abelian difference sets with multiplier −1’, Designs, Codes and Cryptography 1 (1992), 321–322.CrossRefGoogle Scholar
  11. [11]
    R.L. McFarland ’A family of difference sets in non-cyclic groups’, J. Combin. Theory Ser. A 15 (1973), 1–10.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    W. M. Schmidt, Diophantine Approximations, Springer Verlag, LNM 785 (1980).Google Scholar
  13. [13]
    W. M. Schmidt, Diophantine Approximations and Diophantine Equations, Springer Verlag, LNM 1467 (1991).MATHGoogle Scholar
  14. [14]
    N. Tzanakis, J. Wolfskill, ’On the Diophantine equation y2 = 4qn + 4q + 1’, J. Number Theory 23 (1986), 219–237.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    N. Tzanakis, J. Wolfskill, ’The Diophantine equation x2 = 4qα/2+4q+1, with an application to coding theory’, J. Number Theory 26 (1987), 96–116.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag, Basel 2005

Authors and Affiliations

  1. 1.Instituto de Matemáticas, Universidad Nacional Autónoma de MéxicoMorelia, MichoacánMéxico
  2. 2.Department of MathematicsAuburn University MontgomeryMontgomeryUSA

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