Respuesta :
Explanation
By definition, a prime number is a natural number greater than 1 that is not a product of two smaller natural numbers.
We must show that 17 is the only prime number x that can be written as:
[tex]x=n^2-64.[/tex]We can rewrite this expression as:
[tex]x=(n+8)(n-8).[/tex]1) If n = -8 or n = +8, we have:
[tex]x=(n+8)(n-8)=0<1\text{ ^^^^2716},[/tex]2) If -8 < n < 8, we have:
[tex]n+8>0\text{ and }n-8<0\Rightarrow x<0<1\text{ ^^^^2716}[/tex]3) If n < -8 or n > 8, we have:
[tex]x=(n+8)(n-8)>0\text{ \checkmark}[/tex]4) By assumming n < -8 or n > 8, x is a primer number only if (n + 8) or (n - 8) is a ±1.
For both cases, we have:
[tex]\begin{gathered} \text{If }n+8=1\Rightarrow n=-7\Rightarrow(n-8)=-15\Rightarrow x=(n+8)(n-8)=-15<0\text{ ^^^^2716} \\ \text{If }n+8=-1\Rightarrow n=-7\Rightarrow(n-8)=-1\Rightarrow x=(n+8)(n-8)=17>0\text{ }✓. \end{gathered}[/tex]So the only possibility is to have x = 17.
5) We know that 17 is a prime because it is only divisible by 1 and 17.
AnswerWe rewrite the expression as:
[tex]x=(n+8)(n-8).[/tex]1) If n = -8 or n = +8, we have:
[tex]x=(n+8)(n-8)=0<1\text{ ^^^^2716},[/tex]2) If -8 < n < 8, we have:
[tex]n+8>0\text{ and }n-8<0\Rightarrow x<0<1\text{ ^^^^2716}[/tex]3) If n < -8 or n > 8, we have:
[tex]x=(n+8)(n-8)>0\text{ \checkmark}[/tex]4) By assumming n < -8 or n > 8, x is a primer number only if (n + 8) or (n - 8) is a ±1.
For both cases, we have:
[tex]\begin{gathered} \text{If }n+8=1\Rightarrow n=-7\Rightarrow(n-8)=-15\Rightarrow x=(n+8)(n-8)=-15<0\text{ ^^^^2716} \\ \text{If }n+8=-1\Rightarrow n=-7\Rightarrow(n-8)=-1\Rightarrow x=(n+8)(n-8)=17>0\text{ }✓. \end{gathered}[/tex]So the only possibility is to have x = 17.
5) We know that 17 is a prime because it is only divisible by 1 and 17.
