TITLE: Smallest Prime-Factor of $4^{52} + 52^{2013} + 2013^{52}$
QUESTION [6 upvotes]: I would like to find out the smallest prime factor of $4^{52} + 52^{2013} + 2013^{52}$ by hand.
Thanks in advance

REPLY [2 votes]: As I've commented, the divisibility of $4^{52} + 52^{2013} + 2013^{52}$ by $7$ can be derived very easily as follows
$\displaystyle52\equiv3\pmod7\implies52^{2013}\equiv3^{2013}$
Now as $\displaystyle3^3=27\equiv-1\pmod7$ and $\displaystyle2013=3\cdot\text{ some odd number (namely}, 671 ),$
$\displaystyle 3^{2013}\equiv(-1)^{\text{ some odd number }}\pmod7\equiv-1\ \ \ \ (1)$
Again as $2013\equiv4\pmod 7\implies 2013^{52}\equiv4^{52},$
$\displaystyle 4^{52}+2013^{52}\equiv2\cdot4^{52}\pmod7\equiv2\cdot2^{104}=2^{105}=(2^3)^{35}\equiv1^{35}\pmod7$ as $2^3=8\equiv1\pmod7$
$\displaystyle\implies4^{52}+2013^{52}\equiv1\pmod7\ \ \ \ (2)$