Prove that $3^n+2$ is not a prime for all $n\ge 0$, $n$ is a whole number.

Question

Prove that $3^n+2$ is not a prime for all $n \geq 0$, $n$ is a whole number.

The only proof that I can think of (which is not exactly the right way), would be to put $n=0, n=1, \dots, n=5$, and at $n=5$ we get a number that's not prime. Hence we have proved that $3^n+2$ is not prime.

Answer 1

To prove the given statement it is sufficient to find a natural number $n\geq0$ such that $3^n+2$ is a prime number. Take $n=0$. Then $3^0+2=1+2=3$ is prime. Perhaps a more interesting result to prove could be to show that $3^n+2$ is prime for some $n\in\mathbb{N}^+$.

Answer 2

Note that $(\Bbb Z/5 \Bbb Z)^{\ast}$ is cyclic of order $4$ so $\forall m \in \Bbb N~(3^{4m+1} \equiv 3 \pmod 5)$, which means $\forall m \in \Bbb N~(5 \mid (3^{4m+1}+2))$. Thus, there are infinitely many natural numbers $n$ such that $3^n+2$ is composite.