On digits of Mersenne numbers

• Bryce Kerr ^[1] ; László Mérai ^[2] ; Igor E. Shparlinski ^[3]
  1. [1] Max Planck Institute for Mathematics

     Max Planck Institute for Mathematics

     Kreisfreie Stadt Bonn, Alemania

     
  2. [2] Johann Radon Institute for Computational and Applied Mathematics

     Johann Radon Institute for Computational and Applied Mathematics

     Linz, Austria

     
  3. [3] University of New South Wales

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 38, Nº 6, 2022, págs. 1901-1925
• Idioma: inglés
• DOI: 10.4171/RMI/1316
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• Resumen

  • Motivated by recently developed interest to the distribution of q-ary digits of Mersenne numbers Mp=2p−1, where p is prime, we estimate rational exponential sums with Mp, p≤X, modulo a large power of a fixed odd prime q. In turn this immediately implies the normality of strings of q-ary digits amongst about (logX)3/2+o(1) rightmost digits of Mp, p≤X. Previous results imply this only for about (logX)1+o(1) rightmost digits