On digits of Mersenne numbers

  • Bryce Kerr

    Max Planck Institute for Mathematics, Bonn, Germany
  • László Mérai

    Austrian Academy of Sciences, Linz, Austria
  • Igor E. Shparlinski

    University of New South Wales, Sydney, Australia
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Motivated by recently developed interest to the distribution of qq-ary digits of Mersenne numbers Mp=2p1M_p = 2^p-1, where pp is prime, we estimate rational exponential sums with MpM_p, pXp \le X, modulo a large power of a fixed odd prime qq. In turn this immediately implies the normality of strings of qq-ary digits amongst about (logX)3/2+o(1)(\log X)^{3/2+o(1)} rightmost digits of MpM_p, pXp \le X. Previous results imply this only for about (logX)1+o(1)(\log X)^{1+o(1)} rightmost digits.

Cite this article

Bryce Kerr, László Mérai, Igor E. Shparlinski, On digits of Mersenne numbers. Rev. Mat. Iberoam. 38 (2022), no. 6, pp. 1901–1925

DOI 10.4171/RMI/1316