# Number theory question 2-3

• Oct 18th 2012, 10:02 AM
maximus101
Number theory question 2-3
Show that if $2^{p} -1$ is prime, then $p$ must be prime
• Oct 18th 2012, 10:12 AM
HallsofIvy
Re: Number theory question 2-3
Quote:

Originally Posted by maximus101
Show that if $2^{p} -1$ is prime, then $p$ must be prime

I recommend proving the "contrapositive". Assume p is NOT prime and show that $2^p- 1$ is not prime.

It will help to use $x^n- 1= (x- 1)(x^{n-1}+ x^{n-2}+ x^{n-3}+ \cdot\cdot\cdot+ x+ 1)$. You can't apply that to $2^p- 1$ directly because, taking x= 2, x- 1= 2- 1= 1 so that's not a new factor.
• Oct 26th 2012, 03:51 AM
Salahuddin559
Re: Number theory question 2-3
Another related question, prove that 2^2k - 1 is divisible by 3 always.

Salahuddin
Maths online