Question

Let $p>1$ If $2^p-1$ is prime, prove that pis prime.

Short answer

It is proved that p is prime.

Step-by-step solution

Write the definition of prime

An integer p is known as prime. When $p\ne 0,\pm 1$and role="math" localid="1646248772511" $\pm 1$,$\pm p$would be the only divisors of p.

Show that  p is prime

Assume that p is not prime. Take $p=rs$ with any positive integer $k,s>1$.

This provides as follows:

$$\begin{gathered} 2^p-1={\left(2^k\right)}^s-1 \\ =\left(2^k-1\right)\left(1+2^k+...+2^{k\left(s-1\right)}\right) \end{gathered}$$

So, obtain$2^k-1>0$because$k>1$. This demonstrates that$2^p-1$is not prime.

It is obtained that$2^p-1$is prime then p should be prime from contraposition.

Hence, it is proved that p is prime.