Question

True or false:$n^2-n+11$ is prime for every nonnegative integer n. Justify your answer. [Prime were defined in Exercise 10.]

Short answer

It is true that$n^2-n+11$ is a prime for every non-negative integern.

Step-by-step solution

Consider the given statement for P(n)

Suppose that the following property $P\left(n\right)$ is as follows:

$P\left(n\right)=n^2-n+11$

 Check the equation for n = 1

Substitute 1 for n in the equation $P\left(n\right)=n^2-n+11$as follows:

$\begin{array}{rcl}P\left(1\right)& =& {\left(1\right)}^2-\left(1\right)+11\\ & =& 1-1+11\\ & =& 11\end{array}$

Substitute 2 for n in the equation $P\left(n\right)=n^2-n+11$as follows:

$\begin{array}{rcl}P\left(2\right)& =& {\left(2\right)}^2-\left(2\right)+11\\ & =& 4-2+11\\ & =& 13\end{array}$

Check the equation for n = k 

Substitute k for n in the equation $P\left(n\right)=n^2-n+11$ as follows:

$P\left(k\right)=k^2-k+11$

Suppose$P\left(k\right)$is prime for every non-negative integerk.

Check the equation for n = k+1 

Substitutek+1 forn in the equation$P\left(n\right)=n^2-n+11$ as follows:

$\begin{array}{rcl}P\left(k+1\right)& =& {\left(k+1\right)}^2-\left(k+1\right)+11\\ & =& k^2+1+2k-k-1+11\\ & =& k^2+k+11\end{array}$

$P\left(k+1\right)=k^2+k+11$represents a prime number. So, it is true for every non-negative integern.