Even though n is a prime number, 2n - 1 may still not be prime.

The first example you have seen is 211 - 1.

Here is another example, 223 - 1.

Find a prime factor of this number. You may work it out with the help of
a calculator, and the table of prime numbers (< 100) below:

It probably took you quite a while to find its prime factor.The only other
prime factor is quite big.

Here it is:

223 - 1 = 8388607 = 47 x 178481