On digits of Mersenne numbers
Bryce Kerr, L\'aszl\'o M\'erai, Igor E. Shparlinski

TL;DR
This paper studies the distribution of q-ary digits in Mersenne numbers, providing new estimates on exponential sums that imply a higher degree of normality in their digit strings than previously known.
Contribution
It introduces improved bounds on rational exponential sums for Mersenne numbers, leading to stronger results on the normality of their q-ary digit distribution.
Findings
Demonstrates normality of about ()^{3/2+o(1)} rightmost digits of Mersenne numbers.
Provides new estimates on exponential sums involving Mersenne numbers.
Extends previous results from ()^{1+o(1)} to ()^{3/2+o(1)} digits.
Abstract
Motivated by recently developed interest to the distribution of -ary digits of Mersenne numbers , where is prime, we estimate rational exponential sums with , , modulo a large power of a fixed odd prime . In turn this immediately implies the normality of strings of -ary digits amongst about rightmost digits of , . Previous results imply this only for about rightmost digits.
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Taxonomy
TopicsAnalytic Number Theory Research · Algebraic Geometry and Number Theory · Limits and Structures in Graph Theory
