Question

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

Answer 1

$x^2-x+41$

$P\left(n\right)=x^2-x+41$ is a prime number

$P\left(1\right)=1^2-1+41$ is a prime No. of

$P\left(1\right)=41$ is prime, which is true

$P\left(2\right)=2^2-2+41$ is a prime No, i.e.,

$P\left(2\right)=43$ is prime No, which is true

$P\left(211\right)={41}^2-41+41$ is a prime No,

i.e., $P\left(41\right)={41}^2$ is a primes which is Not true.