Advertisement

Resonance

, Volume 23, Issue 7, pp 749–758 | Cite as

On a Problem of Alaoglu and Erdős

  • K. Senthil Kumar
  • R. Thangadurai
  • Veekesh Kumar
Article
  • 8 Downloads

Abstract

Starting with an elementary problem that appeared in the Putnam mathematics competition, we proceed to discuss some techniques of transcendental number theory and prove the following result. If p, q, r are distinct primes and if c is a real number with the property that pc, qc, rc are integers, then c must be a non-negative integer. The tools used are some linear algebra and complex analysis. The zero-density estimate method discussed here was used by Alan Baker to prove his celebrated theorem on linear forms in logarithms. The question as to whether we can replace three primes by two primes is an open question.

Keywords

Six exponentials theorem Siegel’s lemma zero-density estimate transcendental method 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Suggested Reading

  1. [1]
    K Ramachandra, Contributions to the Theory of Transcendental Numbers. I, II, Acta Arith., 14, pp.65–72, pp.73–88, 1967–68.CrossRefGoogle Scholar
  2. [2]
    S Lang, Introduction to Transcendental Numbers, Addison–Wesley, 1966.Google Scholar
  3. [3]
    L Alaoglu and P Erdős, On Highly Composite and Similar Numbers, Trans. Amer. Math. Soc., 56, pp.448–469, 1944.CrossRefGoogle Scholar
  4. [4]
    C L Siegel, Über Einige Anwendungen Diophantischer Approximationen, Abh. Preuss. Akad. Wiss. Phys. Math, pp.41–69. Reprinted in Gesammelte Abhandlungen, Volume 1, 1929.Google Scholar
  5. [5]
    S Lang, Complex Analysis, GTM 103, Fourth Edition, Springer–Verlag, 1999.CrossRefGoogle Scholar
  6. [6]
    K Senthil Kumar, R Thangadurai and M Waldschmidt, Liouville Numbers and Schanuel’s conjecture, Arch. Math., 102, pp.59–70, 2014.CrossRefGoogle Scholar
  7. [7]
    M Ram Murty, Problems in Analytic Number Theory, GTM 206, Readings in Mathematics, Springer, New York, 2008.Google Scholar
  8. [8]
    A Baker, Transcendental Number Theory, Cambridge University Press, 1975.CrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  • K. Senthil Kumar
    • 1
  • R. Thangadurai
    • 2
  • Veekesh Kumar
    • 2
  1. 1.National Institute of Science Education and ResearchHBNIP.O. Jatni, KhurdaIndia
  2. 2.Harish-Chandra Research InstituteHBNIJhunsi AllahabadIndia

Personalised recommendations