Ex 9.1, 14

Transcript

Ex 9.1, 14 Find counterexamples to the following statements. If n is a prime number, then 2ⁿ − 1 is a prime number. If n is an even number, then 2ⁿ + 1 is a prime number. Let’s do it one-by-one If 𝒏 is a prime number, then 𝟐^𝒏−𝟏 is a prime number. Take 𝒏=𝟏𝟏, which is prime Now, 𝟐^𝟏𝟏−𝟏=2048−1 =𝟐𝟎𝟒𝟕 But: 𝟐𝟎𝟒𝟕=𝟐𝟑 × 𝟖𝟗 Thus, 2047 is composite So 𝑛 is prime, but 2^𝑛−1 is composite ∴ 𝒏=𝟏𝟏 gives a counterexample (ii) If 𝒏 is an even number, then 𝟐^𝒏+𝟏 is a prime number. Take 𝑛=6, which is even Now, 𝟐^𝟔+𝟏=64+1 =𝟔𝟓 But: 65=5 × 13 Thus, 65 is composite So 𝑛 is even, but 2^𝑛+1 is composite. ∴ 𝒏=𝟔 gives a counterexample