View attachment 34746
if any1 could assist me in this would be gr8ly appreciated
before I start, I think I've seen a variant of this question before, but I don't remember using contrapositive? But I think the same use of factorisation should work?
Contrapositive: n is not prime (composite) -> 2^n - 1 is not prime (composite)
If n is composite, there exists a set of integers
such
as n is composite.
Considering LHS
and I'm pretty sure you've seen this type of factorisation somewhere
e.g (you've most likely seen this)
so pretty much the - 1 subtracts all the terms from x*x^{n-2} to x*{1} and you are left with x*x^{n-1} - 1 which is simply x^{n} - 1.
So letting x = 2^j and n = k
Hence clearly
divides
as from the equation, meaning that it has more factors than 1 and itself and therefore
is not prime/composite.
As the contrapositive is logically equivalent to the original statement, if
is prime then
is prime.