Determine (with proof) whether or not there exists a positive integer $N$ such that for every integer $n\ge N$, the number $2^n$ in base ten has two consecutive digits that are equal. (Trailing zeros do not count).
I tested many powers of $2$ for small $n$ and wasn't able to find such an $N$. Can we prove this by contradiction that there does not exist such an $N$? I was thinking that if $N$ does satisfy this, then we can consider certain powers of $2$ that will make a contradiction.