we can write: $displaystyle\integral_{0}^{m}{x^m}\leg\sum_{n=1}^{m}{n^m} \leg\integral_{0}^{m+1}{x^m}$
so $frac{m^{m+1}-(m+1)-mk}}{m+1}\leg\displaystyle\sum_{n=1}^{m}{n^m}\leg\frac{(m+1)^{m+1}-(m+1)-mk}}{m+1}$
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
conqruence is hold for the numbers 1,2,6,42,1806, second one is hold for these numbers