isi-entrance 2006 Q3

isi-entrance · India · solved Proof Proof Involving Combinatorial or Number-Theoretic Structure
Show that $n^4 + 4^n$ is composite for all integers $n > 1$.
Here $n$ must be odd $n = 2k+1$. Now, $n^4 + 4^n = n^4 + 4 \cdot 4^{2k} = n^4 + 4 \cdot (2^k)^4$. From Sophie Germain identity: $a^4 + 4b^4 = (a^2 + 2b^2 + 2ab)(a^2 + 2b^2 - 2ab)$, we are done. 
Show that $n^4 + 4^n$ is composite for all integers $n > 1$.
Paper Questions
Q1 Q2 Q3 Q4 Q5 Q7 Q8 Q9 Q10