Question

$\mathbf{\text{Use Exercise}}\mathbf{41}\mathbf{\text{to determine whether}}{\mathit{M}}_{\mathbf{13}}\mathbf{=}{\mathbf{2}}^{\mathbf{13}}\mathbf{-}\mathbf{1}\mathbf{=}\mathbf{8191}\mathbf{\text{and}}{\mathit{M}}_{\mathbf{23}}\mathbf{=}{\mathbf{2}}^{\mathbf{23}}\mathbf{-}\mathbf{1}\mathbf{=}\mathbf{8}\mathbf{,}\mathbf{388}\mathbf{,}\mathbf{607}\mathbf{\text{are prime.}}$

Short answer

$$\begin{gathered} {\mathit{M}}_{\mathbf{13}}\mathbf{=}{\mathbf{2}}^{\mathbf{13}}\mathbf{-}\mathbf{1}\mathbf{=}\mathbf{8191}\mathbf{}\mathrm{is}\mathrm{a}\mathrm{prime} \\ {\mathit{M}}_{\mathbf{23}}\mathbf{=}{\mathbf{2}}^{\mathbf{23}}\mathbf{-}\mathbf{1}\mathbf{=}\mathbf{8}\mathbf{,}\mathbf{388}\mathbf{,}\mathbf{607}\mathbf{}\mathrm{i}\mathrm{s}\mathrm{not}\mathrm{a}\mathrm{prime} \end{gathered}$$

Step-by-step solution

 To check whether M13=213−1=8191 is prime or not 

$\text{From exercise}41\text{we know that everv divisor of}{\mathbf{m}}_{13}\text{is of the form}\mathbf{26}\mathit{k}\mathbf{+}\mathbf{1}\text{with}\mathit{k}$$\text{non-negative.}$

$$\begin{gathered} \text{Lets try different values of}\mathbf{k} \\ \text{Let}\mathbf{k}\mathbf{=}\mathbf{1} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{8191}\mathbf{=}\mathbf{27}\mathbf{\times }\mathbf{303}\mathbf{+}\mathbf{10} \end{gathered}$$

Here $\mathbf{53}$is not a divisor

Let$\mathit{k}\mathbf{=}\mathbf{3}$

$\mathbf{8191}\mathbf{=}\mathbf{79}\mathbf{\times }\mathbf{103}\mathbf{+}\mathbf{54}$

Here $\mathbf{79}$is not a divisor

Let localid="1668682166065" $\mathit{k}\mathbf{=}\mathbf{3}$

$\mathbf{8191}\mathbf{=}\mathbf{79}\mathbf{\times }\mathbf{103}\mathbf{+}\mathbf{54}$

Here localid="1668682241864" $\mathbf{105}$is a divisor

Hencelocalid="1668682338704" $\mathbf{8191}$is a prime