Let $n=3^{1000}+1$. Is n prime?

Question

Let $n=3^{1000}+1$. Is n prime?

My working so far:

$n=3^{1000}+1 \cong 1 \mod 3$

I notice that n is of form; $n=3^n+1$

Seeking advice tips, and methods on progressing this.

Answer 1

Since we have

$$3\equiv 1 \pmod 2 \implies 3^{1000} \equiv 1 \pmod 2 \implies 3^{1000}+1\equiv 0\pmod 2$$

$3^{1000}+1$ is not a prime.

Answer 2

Taking binomial expansion,

\begin{align} 3^{1000}+1&=(2+1)^{1000}+1\\ &=1+\sum_{k=0}^{1000}{1000\choose k}2^k1^{1000-k}\\ &=1+{1000\choose 0}+\sum_{k=1}^{1000}{1000\choose k}2^k\\ &=2\left[1+\sum_{k=1}^{1000}{1000\choose k}2^{k-1}\right] \end{align}

So $3^{1000}+1$ is composite.