TITLE: None of $3,5,7$ can divide $r^4+1$
QUESTION [9 upvotes]: Let $n=r^4+1$ for some $r$. Show that none of $3,5,$ and $7$ can divide $n$.
I am thinking to use a corollary that "each prime divisor p of an integer of the form $(2m)^4+1$ has the form $8k+1$", but I failed. Anyone can give some hint?

REPLY [2 votes]: Check for $r=3k,3k+1,3k+2$
Then check for $r=5k,5k+1,5k+2,5k+3,5k+4$
Then check for $r=7k,7k+1,7k+2,7k+3,7k+4,7k+5,7k+6$
It is the easiest way but lengthy.