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

On a Problem of Alaoglu and Erdős

  • K. Senthil KumarEmail author
  • R. Thangadurai
  • Veekesh Kumar


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.


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


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Suggested Reading

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Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  • K. Senthil Kumar
    • 1
    Email author
  • 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

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