To determine the values of $n$ for which the expression $n^2 + 2023n + 2024$ is not a prime number, we need to find values of $n$ that make the expression composite (i.e., not prime).

The expression given is: $P(n) = n^2 + 2023n + 2024$

We will start by checking if this polynomial can be factored for some values of $n$.

1. Factoring the Quadratic Expression: $P(n) = n^2 + 2023n + 2024$ To factor this, we look for two numbers that multiply to $2024$ (the constant term) and add up to $2023$ (the coefficient of $n$).

2. Finding Factors: We observe that: $2023 = 2024 - 1$ Hence, we can rewrite the expression as: $P(n) = n^2 + 2023n + 2024 = (n + 1)(n + 2024)$

3. Analyzing the Factors: The expression $(n + 1)(n + 2024)$ is a product of two integers. For $P(n)$ to be not prime, both factors must be greater than 1 or one of them should be 1 or both negative.

   • If $n + 1 = 1$, then $n = 0$.
   • If $n + 2024 = 1$, then $n = -2023$.

Thus, we found that:

• When $n = 0$, $P(n) = 2024$, which is not a prime.
• When $n = -2023$, $P(n) = 1$, which is not a prime.

Since $n$ can take any integer value and the factors $(n + 1)$ and $(n + 2024)$ must ensure the product is not prime, any value of $n$ that makes $n + 1$ or $n + 2024$ a non-prime value will make $P(n)$ non-prime.

Therefore, $n$ can be $0$ or $-2023$.

Would you like details or have any questions?

Here are 8 related questions to explore further:

1. What makes a number composite?
2. How do we factor quadratic polynomials?
3. Why is $P(n) = n^2 + 2023n + 2024$ factorable as $(n + 1)(n + 2024)$?
4. What other values of $n$ make $n^2 + 2023n + 2024$ not prime?
5. How does the discriminant help in determining the nature of quadratic roots?
6. What is the role of prime and composite numbers in number theory?
7. How can we generalize this factorization approach for other quadratic expressions?
8. What are some efficient algorithms for checking the primality of large numbers?

Tip: Factoring expressions can often reveal simpler ways to identify non-prime values. Always check for factor pairs that sum to the middle coefficient and multiply to the constant term.