# True or False Statement?

• February 11th 2009, 06:51 AM
Edbaseball17
True or False Statement?
If True explain show why and if false give a counterexameple.

For All $n>=1$, if n is not prime, then neither is $2^n-1$.

How would I go about figuring this out? Is there a forumula or do I just plug numbers in?
• February 11th 2009, 08:04 AM
clic-clac
Hi!

A useful formula: $\forall a,b,n\in\mathbb{N},\ n\geq 2,\ a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+...+ab^{n-2}+b^{n-1})$

Let $n=pq$ be a positive integer with $p,q\geq 2$
$2^n-1=2^{pq}-1^q=(2^p)^q-1^q=(2^p-1)(\sum\limits_{k=0}^{q-1}2^k)$

Hence $\forall n\geq 1\ \text{no}(n\ \text{prime})\Rightarrow\ \text{no}(2^n-1\ \text{prime})$