Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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|improve this question
    
hmmm... you may want to look up Poincare's recurrence theorem: en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theorem –  WWright Dec 5 '10 at 21:01
add comment

1 Answer

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|improve this answer
3  
You don't need equidistribution; this follows from Dirichlet's approximation theorem (which is the reason the pigeonhole principle is named after Dirichlet): en.wikipedia.org/wiki/Dirichlet's_approximation_theorem –  Qiaochu Yuan Dec 6 '10 at 16:35
add comment

Your Answer

 
discard

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