<s>[INST] On prime numbers let $q$ be a prime 
let $p = 2^q -1 $
is p must be prime always for any prime q ? 
is this is true always ? 
or it is false for some prime q ? 
if it is false , give an example to show that there is a prime q such that 
$2^q -1$ is not a prime 
thanx 
 [/INST] Here's some example computations to suggest a way you might try to prove that $q$ must be prime:
$$x^2 - 1 = (x - 1)(x + 1)$$
So:
$$x^6 - 1 = (x^3)^2 - 1 = (x^3 - 1)(x^3 + 1)$$
And so:
$$2^6 - 1 = 63 = (2^3 - 1)(2^3 + 1) = 7 \cdot 9$$
Or:
$$x^3 - 1 = (x - 1)(x^2 + x + 1)$$
So:
$$x^{15} - 1 = (x^5)^3 - 1 = (x^5 - 1)(x^{10} + x^5 + 1)$$
And so:
$$2^{15} - 1 = 32767 = (2^5 - 1)(x^{10} + 2^5 + 1) = 31 \cdot 1057$$
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