show this number maybe is prime?

Question

prove or disprove $$2016^{2017}+1008^{2017}\cdot 2017^{1008}+(2017)^{2016}$$ not prime number?

It's probably based on factorization. $2016=1008\cdot 2$

Answer 1

Fermat's little theorem gives us that $a^{p-1}\equiv 1 \bmod p$ for $p$ prime and $p\nmid a$. In particular $a^4\equiv 1\bmod 5$ for $5\nmid a$

$\begin{align} 2016^{2017}+1008^{2017}\cdot 2017^{1008} &+(2017)^{2016} \\ &\equiv 1^{2017}+3^{2017}\cdot 2^{1008}+2^{2016} \bmod 5 \\ &\equiv 1+3^{1}\cdot 1^{504}\cdot 1^{252}+1^{504} \bmod 5 &\text{Fermat's little thm}\\ &\equiv 5 \bmod 5 \\ \end{align}$