Question

(a) Verify that $2^5-1$and $2^7-1$are prime.

(b) Show that$2^{11}-1$ is not a prime.

Short answer

1. It is verified that$2^5-1$ and$2^7-1$ are prime.
2. It is proved that$2^{11}-1$ is not a prime.

Step-by-step solution

Verify that 25-1 is prime

Find the value of$2^5-1$ asfollows:

$\begin{array}{rcl}{2}^5-1& =& 32-1\\ & =& 31\end{array}$

As $\sqrt{31}<6$, the number 31 is not divided by primes numbers 2, 3, and 5. This implies that 31 is a prime number.

Hence, it is verified thatrole="math" localid="1646127324163" $2^5-1$ is prime.

Verify that 27-1is prime

Find the value of $2^7-1$as follows:

$\begin{array}{rcl}{2}^7-1& =& 128-1\\ & =& 127\end{array}$

As $\sqrt{127}<12$,the number 127 is not divided by primes numbers 2, 3, 5, 7,and 11. This implies that 127 is a prime number.

Hence, it is verified that$2^7-1$ is prime.

Prove that 211-1 is prime

Find the value of $2^{11}-1$as follows:

$\begin{array}{rcl}{2}^{11}-1& =& 2048-1\\ & =& 2047\\ & =& 23·89\end{array}$

As the numbers 23 and 89 are prime. This implies that$2^{11}-1$ is a prime number.

Hence, it is proved that$2^{11}-1$ is prime.