The trend of huge number (3)

Relevant wiki: Modular Arithmetic - Exponentiation

Simple solution :

$2016^{{1}^{2017}}+2016^{{2}^{2017}}+2016^{{3}^{2017}}+\dots+2016^{{2016}^{2017}}\equiv x \pmod{2017}$

$(-1)^{{1}^{2017}}+(-1)^{{2}^{2017}}+(-1)^{{3}^{2017}}+\dots+(-1)^{{2016}^{2017}}\equiv x \pmod{2017}$

$(-1)+1+(-1)+\dots+1 \equiv \pmod{2017}$

$x \equiv 0 \pmod{2017}$

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Complicated solution :

$2016^{{1}^{2017}}+2016^{{2}^{2017}}+2016^{{3}^{2017}}+\dots+2016^{{2016}^{2017}}\equiv x \pmod{2017}$

$2016^{1}+2016^{2}+2016^{3}+\dots+2016^{2016}\equiv x \pmod{2017}$ (By Fermat's little theorem)

$\frac{2016(2016^{2016}-1)}{2016-1}\equiv x \pmod{2017}$

$\frac{(-1)((-1)^{2016}-1)}{2015}\equiv x \pmod{2017}$

$\frac{(-1)(1-1)}{2015}\equiv x \pmod{2017}$

$x \equiv 0 \pmod{2017}$