Perfect powers with few binary digits and related Diophantine problems

• Michael A. Bennett ^[1] ; Yann Bugeaud ^[2] ; Maurice Mignotte ^[2]
  1. [1] University of British Columbia

     University of British Columbia

     Canadá

     
  2. [2] University of Strasbourg

     University of Strasbourg

     Arrondissement de Strasbourg-Ville, Francia

     
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 12, Nº 4, 2013, págs. 941-953
• Idioma: inglés
• Enlaces
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• Resumen

  • We prove that, for any fixed base x ≥ 2 and sufficiently large prime q, no perfect q-th power can be written with 3 or 4 digits 1 in base x. This is a particular instance of rather more general results, whose proofs follow from a combination of refined lower bounds for linear forms in Archimedean and non- Archimedean logarithms.