Question

If P(n) is the statement "$n^2-n+41$ is prime", prove that P(1), P(2) and P(3) are true. Prove also that P(41) is not true.

Answer 1

P(n) : $n^2-2+41$ is prime

P(1) : 1 - 1 + 41 is prime

$\Rightarrow P\left(1\right)+41$ is prime

$\therefore$ P(1) is true.

P(2) : $2^2-2+41$ is prime

$\Rightarrow$ P(2) : 43 is prime

$\therefore$ P(2) is true.

P(3) : $3^2-3+41$ is prime

$\Rightarrow$ P(3) : 47 is prime

$\therefore$ P(3) is true.

P(41) : $(41{)}^2-41+41$ is prime

P(41) : $(41{)}^2$ is prime

$\Rightarrow$ P(41) is not ture