Question

assume: n is a positve integer, p is a prime number if 2n-1 = p show that n is also prime

Answer 1

hmmm . 'n' is not always prime for certain values of p . it works for when p <= 13 .

Answer 2

Did you mean \(2^n-1\)? As MrHoola said, a simple counter example is p = 17, n = 9.

Answer 3

yeah \[2^{n}-1 = p\]

Answer 4

Assume \(n \) is not prime, then \(n = ab\) implying that:
\[2^n-1 = 2^{ab}-1 = (2^a)^b-1\]
We can factor \((2^a)^b-1\) as:
\[ p= ((2^a)^b-1) = (2^a-1)((2^a)^{b-1} +(2^a)^{b-2} +...+1)\]
Note that this implies that \(p\) is a product of 2 numbers and not prime.