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Total digit sum in distinct bases grow unboundedlly
For positive integers $n$ and $d\ge 2$, let $S_d(n)$ be the digit sum of $n$ in base $d$.
When $a$ and $b$ are not powers of each other, is it always true that
$$\liminf_{n\to\infty} S_a(n)+S_b(n)=\in …