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Question -

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.



Answer -

Given:

P(n) = n2 – n + 41 is prime.

P(n) = n2 – n + 41

P (1) = 1 – 1 + 41

= 41

P (1) is Prime.

Similarly,

P(2) = 22 – 2 + 41

= 4 – 2 + 41

= 43

P (2) is prime.

Similarly,

P (3) = 32 – 3 + 41

= 9 – 3 + 41

= 47

P (3) is prime

Now,

P (41) = (41)2 – 41 + 41

= 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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