we can write:
$\displaystyle\int_{0}^{m}{x^m}\le\sum_{n=1}^{m}{n^m}
\le\int_{0}^{m+1}{x^m}$

so $\displaystyle\frac{m^{m+1}-(m+1)-mk}{m+1}\le\sum_{n=1}^{m}{n^m}\le\frac{(m+1)^{m+1}-(m+1)-mk}{m+1}$

but since $(m+1)^{m+1}\equiv{m+1}\pmod{m}$ and also $m^{m+1}\equiv m\pmod{m}$ so the

 central statement should be zero, or $\displaystyle\sum_{n=1}^{m}{n^m}$, when the first

 congruence holds for the numbers 1,2,6,42,1806, second one holds for these numbers