Digit Occurrence in Geometric Progressions

Digit Occurrence in Geometric Progressions

Mathematics – Number Theory and Dynamical Systems

21/09/2026

Joel Moreira

This project investigates digit phenomena motivated by a question of Paul Erdős. We study the occurrence of the digit 7 in powers of 2, asking whether every sufficiently large power of 2 contains the digit 7. More generally, we consider the sequence (a·2ⁿ) for varying positive integers a and formulate the following conjecture: the set of a for which a·2ⁿ contains the digit 7 for all sufficiently large n has full natural density. In pursuit of this conjecture, we develop useful combinatorial and arithmetic estimates, together with a dynamical approach to the leading digits of these numbers.

number theory, digit occurrence, powers of two, geometric progressions, Benford's law, natural density, logarithmic density, dynamical systems, shrinking targets, irrational rotations, combinatorics, Erdős