Ex 9.1, 13

Transcript

Ex 9.1, 13 There are no known ‘neat’ expressions that generate only primes! Find counterexamples to the following claims. All numbers of the form 4n² + 1 are prime. All numbers of the form n² + n + 11 are prime. All numbers of the form 4n + 3 are prime. For each claim, choose a value of 𝒏 that produces a composite number. (i) All numbers of the form 𝟒𝒏^𝟐+𝟏 are prime Take 𝒏=𝟒 Now, 𝟒𝒏^𝟐+𝟏=4(4^2 )+1 =4(16)+1 =𝟔𝟓 But: 65 = 5 × 13 Thus, 65 is composite ∴ The claim is false; 𝒏=𝟒 gives a counterexample (ii) All numbers of the form 𝒏^𝟐+𝒏+𝟏𝟏 are prime Take 𝒏=𝟏𝟎 Now, 𝒏^𝟐+𝒏+𝟏𝟏=10^2+10+11 =100+10+11 =𝟏𝟐𝟏 But: 121 = 11 × 11 Thus, 121 is composite ∴ The claim is false; 𝒏=𝟏𝟎 gives a counterexample (iii) All numbers of the form 𝟒^𝒏+𝟑 are prime Take 𝒏=𝟓 Now, 𝟒^𝒏+𝟑=4^5+3 =1024+3 =𝟏𝟎𝟐𝟕 But: 1027 = 13 × 79 Thus, 1027 is composite ∴ The claim is false; 𝒏=𝟓 gives a counterexample