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Problem:

For positive integers $n,k$, let $$S(n,k)=\sum_{i=1}^{n}i^k$$ and for positive integers $m,b$, with $b>1$, let $D(m,b)$ be the sum of the base-$b$ digits of $m$.

Q$1$- Show that if $k\in\{1,2,3\}$, and $a$ is a positive integer such that $a{\,|\,}S(a,k)$, then $D(S(b,k),b)=b$, where $b=a+1$ but not satisfied $\forall$ $k > 3$?

Q$2$- Show that $D((p')^{t}-D((p')^{2k+1}-S(p',2k),p'),p')=p'$ where $p$ is prime and $p+1=p'$ and $p>2k+1$ and $(p')^{t} \ge D((p')^{2k+1}-S(p',2k),p')>(p')^{t-1}$ and $k,t \in \mathbb{N}$?

note

Quasi well proved half of question$1$ in this link proof for $k\in\{1,2,3\}$

I have already mention the observation of question$2$ in different manner in this link reference for Q$2$

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