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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.
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Solution
\[P(n): n^2 - n + 41\text{ is prime } . \]
\[Now, \]
\[P(1) = 1^2 - 1 + 41 = 41 (\text{ prime} )\]
\[P(2) = 2^2 - 2 + 41 = 4 - 2 + 41 = 43 (\text{ prime} )\]
\[P(3) = 3^2 - 3 + 41 = 9 - 3 + 41 = 47 (\text{ prime } )\]
\[P(41) = {41}^2 - 41 + 41 = 1681 (\text{ not prime} )\]
\[\text{ Thus, we can say that P(1), P(2) and P(3) are true, but P(41) is not true } .\]
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