Sets, Relations & Functions
Principle of Mathematical Induction
Grade None
Question:
<p>Given \(P(n) = x^2 - n + 41\) is prime. Which of the following options is correct regarding \(P(3)\) and \(P(5)\)?</p>
<p>(1) Both \(P(3)\) and \(P(5)\) are true</p>
<p>(2) \(P(3)\) is true but \(P(5)\) is false</p>
<p>(3) \(P(3)\) is false but \(P(5)\) is true</p>
<p>(4) Both \(P(3)\) and \(P(5)\) are false</p>
Step-by-Step Solution
Key Concept: Substitute the given values directly into the polynomial and check primality by testing divisibility or recognizing the resulting number as prime or composite.
<p><strong>Step 1:</strong> Calculate P(3)</p><p>P(3) = 3² - 3 + 41 = 9 - 3 + 41 = 47</p><p>Check: 47 is prime (not divisible by 2, 3, 5, or 7)</p><p><strong>Step 2:</strong> Calculate P(5)</p><p>P(5) = 5² - 5 + 41 = 25 - 5 + 41 = 61</p><p>Check: 61 is prime (not divisible by 2, 3, 5, or 7)</p><p><strong>Step 3:</strong> Verify observations</p><p>Both P(3) = 47 and P(5) = 61 are prime numbers. This polynomial generates primes for many initial values, but fails at n = 40 where P(40) = 40² - 40 + 41 = 1681 = 41²</p><p>∴ Answer: A</p>
Correct Answer: A