Consider the statement: "$\mathrm{P}(\mathrm{n}): \mathrm{n}^{2}-\mathrm{n}+41$ is prime." Then which one of the following is true?

(A) Both $\mathrm{P}(3)$ and $\mathrm{P}(5)$ are true

(B) $\mathrm{P}(3)$ is false but $\mathrm{P}(5)$ is true

(C) Both $P(3)$ and $P(5)$ are false

(D) $P(5)$ is false but $P(3)$ is true

JEE Main · 2019 · 10 Jan 2019 · Shift 1 · Mathematical Induction

Consider the statement: "

P (n) : n^{2} - n + 41

is prime." Then which one of the following is true?

Answer: (A)

Testing the claim directly,

P (3) = 9 - 3 + 41 = 47

, which is prime, and

P (5) = 25 - 5 + 41 = 61

, which is also prime. Both statements therefore hold. The polynomial

n^{2} - n + 41

is famous for staying prime for small

n

, though it does eventually fail, just not at

n = 3

5

JEE Main · 2019 · 10 Jan 2019 · Shift 1 · Single correct option