Proof：練習問題

Which of the following integers provides a counter-example to disprove the statement: "For all positive integers \(n\), \(n^2 + n + 41\) is a prime number"?

Which of the following values of \(p\) serves as a counter-example to disprove the statement: "For every prime number \(p\), the value \(p + 2\) is also a prime number"?

Consider the statement: "For all positive integers \( n \), \( n^2 - n + 41 \) is a prime number." Which of the following values of \( n \) provides a counter-example that disproves this claim?

Which of the following statements can be disproved by using the counter-example \(n = 2\)?

Which of the following values of \(n\) serves as a counter-example to disprove the statement: "\(n^2 - n + 11\) is a prime number for all positive integers \(n\)"?

Prove by exhaustion that for all integers \( n \) such that \( 2 \leq n \leq 5 \), the expression \( 2^n - 1 \) is not divisible by 4.

(a) Prove by exhaustion that for all integers \( n \) such that \( 1 \leq n \leq 5 \), the value of \( n^{2} + 2 \) is not divisible by 4.

(b) Prove that the square of any odd integer is always of the form \( 8k + 1 \), where \( k \) is an integer. (Hint: An odd integer can be written as \( 2m + 1 \)).

(c) Disprove the following statement by providing a counter-example: "For all real numbers \( a \) and \( b \), if \( a^{2} = b^{2} \), then \( a = b \)."