Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Prove that for any finite sequence of decimal digits, there exists an $n$ such that the decimal expansion of $2^n$ begins with these digits.

share|cite|improve this question
hmmm... you may want to look up Poincare's recurrence theorem: – WWright Dec 5 '10 at 21:01

Take $\log_{10} (2^n) = n \log_{10} 2$, note that $\log_{10} 2$ is irrational, and use the equidistribution theorem (or prove what you want directly using the pigeonhole principle).

share|cite|improve this answer
You don't need equidistribution; this follows from Dirichlet's approximation theorem (which is the reason the pigeonhole principle is named after Dirichlet):'s_approximation_theorem – Qiaochu Yuan Dec 6 '10 at 16:35

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.