TITLE: Prove a number is composite
QUESTION [14 upvotes]: How can I prove that $$n^4 + 4$$ is composite for all $n > 5$? 
This problem looked very simple, but I took 6 hours and ended up with nothing :(. I broke it into cases base on quotient remainder theorem, but it did not give any useful information.
Plus, I try to factor it out:
$$n^4 - 16 + 20 = ( n^2 - 4 )( n^2 + 4 ) - 5\cdot4$$
If a composite is added to a number that is a multiple of $5$, is there anything special? A hint would suffice.
Thanks,
Chan

REPLY [4 votes]: $$ x^4+4=\\
[(x^2)^2+4x^2+4]-4x^2\\
=(x^2+2)^2-(2x)^2\\
=(x^2+2x+2)(x^2-2x+2)\ldots $$