# Unique pattern in addition of digits

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$$2$$