Let n be an integer greater than 1 and consider the statement “A: $2^n - 1$ prime is necessary for n to be prime”. I need to prove A is true or false and am having difficulty with the proof.
Also, a hint is given that $(2^a)^b - 1 = (2^a - 1)[(2^a)^{b-1} + (2^a)^{b-2} +\ldots+ 2^a + 1]$.
I believe the hint may be involved in the proof of statement A but I’m not sure how to use this hint in the proof.
In order to prove this equality $$(2^a)^b - 1 = (2^a - 1)[(2^a)^{b-1} + (2^a)^{b-2} +\ldots+ 2^a + 1]$$ we let $$x=2^a$$ Thus $$x^b - 1 = (x - 1)[x^{b-1} + x^{b-2} +\ldots+ x + 1]$$
which is well-known and could be easily verified by multiplication.