Disprove the following statement by giving a counterexample.
For every integer p, if p is prime then 
p2 − 1
 is even.
Counterexample: Consider the ordered pair 
(p, p2 − 1) = 


We are tasked with disproving the statement: 

"For every integer \( p \), if \( p \) is prime, then \( p^2 - 1 \) is even."

### Counterexample:
Consider the prime number \( p = 2 \).

1. \( p^2 = 2^2 = 4 \)
2. \( p^2 - 1 = 4 - 1 = 3 \)

Here, \( 3 \) is an odd number, which contradicts the claim that \( p^2 - 1 \) is even for all primes \( p \).

Thus, the statement is false. A counterexample is the ordered pair \( (2, 3) \), where \( p = 2 \) and \( p^2 - 1 = 3 \), which is odd.

Would you like more details on this or have any questions?

Here are 5 follow-up questions to further explore related concepts:
1. What are the conditions for a number to be considered prime?
2. Can you generalize a statement about when \( p^2 - 1 \) is even?
3. How does the parity (odd/even nature) of numbers change after squaring?
4. What other small primes might provide insight into this problem?
5. Is \( p^2 - 1 \) always odd for certain types of primes?

**Tip**: When analyzing patterns involving primes, remember that 2 is the only even prime number, which often leads to unique cases.

Disprove the following statement by giving a counterexample. For every integer p, if p is prime then p^2 − 1 is even. Counterexample: Consider the ordered pair (p, p^2 − 1) = 

Disprove the statement 'If p is prime, then p^2 - 1 is even' by providing a counterexample. Using the prime number p = 2, we show that p^2 - 1 = 3, which is odd, disproving the statement. Explore key concepts like prime numbers and parity, suitable for students in Grades 8-10.

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