gcd and lcm - Given that $p$ is an odd prime, is the GCD of any two numbers of the form $2^p + 1$ always equal to $3$?

I have checked it for some numbers and it appears to be true. Also I am able to reduce it and get the value $3$ for specific primes $p_1$, $p_2$ by using the Euclidean algorithm but I am not able to find a general argument for all numbers.