Consider the following statement.
There is an integer n such that
2n^2 − 5n + 2
is prime.
To prove the statement it suffices to find a value of n such that
(n, 2n^2 − 5n + 2)
satisfies the property
"2n^2 − 5n + 2
is prime." Show that you can do this by entering appropriate values for n and 2n^2 - 5n + 2.

To prove the statement that there exists an integer \( n \) such that \( 2n^2 - 5n + 2 \) is prime, we need to find a specific value of \( n \) for which \( 2n^2 - 5n + 2 \) evaluates to a prime number.

Let's check a few small integer values for \( n \):

### 1. \( n = 1 \):
\[
2(1)^2 - 5(1) + 2 = 2(1) - 5 + 2 = 2 - 5 + 2 = -1 \quad \text{(not prime)}
\]

### 2. \( n = 2 \):
\[
2(2)^2 - 5(2) + 2 = 2(4) - 10 + 2 = 8 - 10 + 2 = 0 \quad \text{(not prime)}
\]

### 3. \( n = 3 \):
\[
2(3)^2 - 5(3) + 2 = 2(9) - 15 + 2 = 18 - 15 + 2 = 5 \quad \text{(prime)}
\]

Here, when \( n = 3 \), \( 2n^2 - 5n + 2 = 5 \), and 5 is indeed a prime number.

### Conclusion:
The integer \( n = 3 \) satisfies the condition that \( 2n^2 - 5n + 2 \) is prime. Therefore, this proves the statement.

Would you like more details or have any other questions?

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Here are 5 related questions you might consider:

1. What happens if you try other values of \( n \)?
2. Can \( 2n^2 - 5n + 2 \) be prime for negative values of \( n \)?
3. How does changing the coefficients in the quadratic expression affect the result?
4. Are there other integer values of \( n \) for which \( 2n^2 - 5n + 2 \) is prime?
5. What other types of expressions can yield prime numbers?

**Tip:** Always check small values first when trying to prove a statement with an example, as they often reveal patterns or solutions quickly.

Explore how to prove the statement that there exists an integer n such that 2n^2 - 5n + 2 is prime. Check detailed examples and solutions for various values of n. Suitable for high school students and math enthusiasts.

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