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.