If P (n) is the statement “n2 – n + 41 is prime”, prove that P (1), P (2) and P (3) are true. Prove also that P (41) is not true.

Home » If P (n) is the statement “n2 – n + 41 is prime”, prove that P (1), P (2) and P (3) are true. Prove also that P (41) is not true.

If P (n) is the statement “n2 – n + 41 is prime”, prove that P (1), P (2) and P (3) are true. Prove also that P (41) is not true.

According to ques:

$P\left( n \right)\text{ }=\text{ }{{n}^{2}}~\text{ }n\text{ }+\text{ }41$

is prime.

$P\left( n \right)\text{ }=\text{ }{{n}^{2}}~\text{ }n\text{ }+\text{ }41$

Or,

$P\text{ }\left( 1 \right)\text{ }=\text{ }1\text{ }\text{ }1\text{ }+\text{ }41$

$=\text{ }41$

P (1) is Prime.

Similarly,

$P\left( 2 \right)\text{ }=\text{ }{{2}^{2}}~\text{ }2\text{ }+\text{ }41$

$=\text{ }4\text{ }\text{ }2\text{ }+\text{ }41$

$=\text{ }43$

P (2) is prime.

Similarly,

$P\text{ }\left( 3 \right)\text{ }=\text{ }{{3}^{2}}~\text{ }3\text{ }+\text{ }41$

$=\text{ }9\text{ }\text{ }3\text{ }+\text{ }41$

$=\text{ }47$

P (3) is prime

Now,

$P\text{ }\left( 41 \right)\text{ }=\text{ }{{\left( 41 \right)}^{2}}~\text{ }41\text{ }+\text{ }41$

$=\text{ }1681$

P (41) is not prime

Hence, P (1), P(2), P (3) are true but P (41) is not true.

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