Any expression that can be written as a difference of two squares can be factorised
using the algebraic identity:

a 2 - b 2 = (a - b) (a + b)

Therefore

214 - 1 = (27 - 1) (27 + 1)

and we have found two factors for M14.

Similarly,

226 - 1 = (213 - 1) (213 + 1)

showing M26 is not a Mersenne prime.

We can conclude that if n is an even number, then Mn will not be a Mersenne prime.
 
 

Here's another problem for you to think about:

Whenever n is an even number, 2n - 1 is divisible by 3.

Can you explain why?