Peter wrote:

Let $S_{n}$ be the sum of the digits of $2^n$ . Prove or disprove that $S_{n+1}=S_{n}$ for some positive integer $n$ .

Let $2^n = a_{m}10^m + a_{m-1}10^{m-1} + ... +a_{1}10 + a_0$
with $a_i$ being integers.
Then $2^n \equiv a_m + a_{m-1} + ... + a_1 + a_0\equiv S(n) (mod3)$
So $S(n+1)\equiv 2^{n+1}\equiv 2.2^n\equiv 2S(n) (mod3)$
So if $S(n+1) = S(n)$ then $S(n)\equiv 0 (mod3)$ and thus $2^n\equiv 0 (mod3)$ which is impossible. Thus $S(n+1)$ cannot equal $S(n)$ for any positive integer $n$ .