I am searching for the earliest published proof of the following result: $$\lim_{k\to\infty} s(2^k) = \infty$$ where $s(n)$ denotes the sum of the decimal digits of $n$.
This problem has been discussed previously on Math.SE. It is proved in the book 250 Problems in Elementary Number Theory (1970) by Wacław Sierpiński, on page 103. The author attributes the proof to Andrzej Schinzel (1937-2021), but he does not provide a reference.
It is known that $\lim_{k\to\infty} s(b^k) = \infty$ for every positive integer $b$, excluding powers of 10. More generally, if $a$ and $b$ are positive integers such that $\log(a)/\log(b)$ is irrational, and if $M$ is a positive integer, then there are only finitely many positive integers whose digital sums in base $a$ and $b$ are both less than $M$. This was proved by H. G. Senge and E. G. Straus in 1973.