How to prove that it exists an integer $n$ such that the decimal expansion of $2^n$ starts with $123456789$?
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I think it can be done by using Pigeon-Hole Principle but I am not sure how can it be done. – ArsenBerk Dec 31 '17 at 17:59
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How does this relate to the dynamical systems tag? – Michael Dec 31 '17 at 18:30
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@Michael The question is from a document studying « les systèmes dynamiques » which is the French translation of « dynamical systems ». And indeed, the question can be looked at the locus of the a particle having the motion $n \mapsto 2^n$ in $\mathbb R$. – mathcounterexamples.net Jan 01 '18 at 06:31