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Problem related to inequality of sum of digits of power sum
Let $D$ be the function define as $D(b,n)$ be the sum of the base-$b$ digits of $n$.
Example: $D(2,7)=3$ means $7=(111)_2\implies D(2,7)=1+1+1=3$
Define $S(a,m)=1^m+2^m+3^m+... …
4
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There are infinitely many prime which have arbitrary large gap in their digits in particular...
Motivation: If Claim 1 is true then we can say there exist arbitrary large gaps between digits of prime $\left[\frac{(mt+r)^m}{m}\right]$ in base expansion $mt+r$, because we can express $\left[\frac{( … frac{(mt+r)^m}{m}\right]=\sum_{i=0}^{m-1}(x_it+y_i)(mt+r)^i$$ for some non negative integer $x_i,y_i$ such that $0\le x_it+y_i<mt+r$ then there exists a positive integer $N$ such that $x_it+y_i$ are digits …