An even bigger prime?

On the Radio 5 Live breakfast show this morning (6^th Feb 2013) Rachel Burden described the newly found biggest prime:  2^57,885,161 - 1    

Nicky Campbell, her co-host on the programme immediately said: “I know a bigger one: how about three to the power of … ”.

This is worth looking at with some classes. 

Nicky Campbell was joking, but how do we know that 3^57,885,161 - 1 isn’t prime?  I like this sort of question because it can be answered if you understand that powers of 3 must all be odd, or you could start with smaller powers: 3² - 1, 3³ - 1, 3⁴ - 1, and discover the answer is always even and could then try to explain it.

How can this be extended:  Well, what must happen to n^57,885,161 - 1 for any odd n (greater than 1)? 

More difficult is looking at some of the even values of n.  n=4 is nice, because we can rewrite it like this:

4^57,885,161 - 1 = (2^57,885,161)² - 1, which is the difference of two squares, so we have 

(2^57,885,161 + 1)(2^57,885,161 - 1) so we have a composite when n=4.

n=6 also works out neatly, because all powers of 6 end in 6, so when we subtract 1 from 6^57,885,161 we have a multiple of 5.

n=8 is harder for KS4 pupils to work on, but if we rewrite as a power of 2 we get something of the form x³ - 1, which factorises to give (x² + x + 1)(x - 1).

n=10 is good too.  We can tell by inspection that it is divisible by 9.  Is it also divisible by 11, though?