MHB Is 2017^4+4^{2017} a Prime Number?

[anemone]

Is $2017^4+4^{2017}$ a prime?

 

[Greg]

Is $2017^4+4^{2017}$ a prime?

Spoiler

$$\begin{align*}2017^4+4^{2017}&=(2017^2)^2+(2^{2017})^2 \\
&=(2017^2+2^{2017})^2-2\cdot2017^2\cdot2^{2017} \\
&=(2017^2+2^{2017})^2-2017^2\cdot2^{2018} \\
&=(2017^2+2^{2017}-2017\cdot2^{1009})(2017^2+2^{2017}+2017\cdot2^{1009})\end{align*}$$

$$Q.E.D.$$

 

[kaliprasad]

Is $2017^4+4^{2017}$ a prime?

Spoiler

$2017^4 + 4^{2017}= (2017^2)^2 + (2^{2017})^2 + 2 * 2017^2 * 2^ {2017} - 2017^2 * ( 2^{1009})^2$
= $(2017^2+ 2^{2017})^2 - (2017 * 2^{1009})^2 $
= $(2017^2+ 2^{2017}+ 2017 * 2^{1009}) (2017^2+ 2^{2017}- 2017 * 2^{1009})$

it not a prime

edit: I posted my solution dispite the fact that gregs' solution came but I did not copy it

 

[anemone]

Thanks both for participating! :D