If P (n) is the statement “n2 – n + 41 is prime”, prove that P

1.	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 as P(n) = n² – n + 41 is prime. P(n) = n² – n + 41 P(1) = 1 – 1 + 41 = 41 P (1) is Prime. Now similarly, P(2) = 2² – 2 + 41 = 4 – 2 + 41 = 43 P (2) is prime. Then similarly, P (3) = 3² – 3 + 41 = 9 – 3 + 41 = 47 P (3) is prime Then, P (41) = (41)² – 41 + 41 = 1681 P (41) is not prime Thus P (1), P(2), P (3) are true but 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.

Answer»

Given as

P(n) = n² – n + 41 is prime.

P(n) = n² – n + 41

P(1) = 1 – 1 + 41

= 41

P (1) is Prime.

Now similarly,

P(2) = 2² – 2 + 41

= 4 – 2 + 41

= 43

P (2) is prime.

Then similarly,
P (3) = 3² – 3 + 41

= 9 – 3 + 41

= 47

P (3) is prime

Then,

P (41) = (41)² – 41 + 41

= 1681

P (41) is not prime

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