Numbers BIG, HCF small

We know that any power of $3$ is odd . $\Rightarrow$ $3^{2014}$ is odd. $\Rightarrow$ $3^{2014}+1$ & $3^{2014}-1$ are consecutive even numbers.

They can be represented as $2m$ & $2n$ where $m$ & $n$ are consecutive numbers. And since any two consecutive numbers are co-prime, $m$ & $n$ are also co-prime.

$\Rightarrow$ the HCF/GCD is $\boxed{2}$ .