What value of “n” will disapprove the statement below?

n²+n+1 is a prime number for all n≥1

The statement suggests that \( n^2 + n + 1 \) is a prime number for all \( n \geq 1 \). To disprove this, we need to find a value of \( n \geq 1 \) such that \( n^2 + n + 1 \) is **not** a prime number.

Let's check a few values of \( n \):

- For \( n = 1 \):
  \[
  n^2 + n + 1 = 1^2 + 1 + 1 = 3 \quad (\text{prime})
  \]

- For \( n = 2 \):
  \[
  n^2 + n + 1 = 2^2 + 2 + 1 = 7 \quad (\text{prime})
  \]

- For \( n = 3 \):
  \[
  n^2 + n + 1 = 3^2 + 3 + 1 = 13 \quad (\text{prime})
  \]

- For \( n = 4 \):
  \[
  n^2 + n + 1 = 4^2 + 4 + 1 = 21 \quad (\text{not prime, since } 21 = 3 \times 7)
  \]

Thus, when \( n = 4 \), the expression \( n^2 + n + 1 \) gives 21, which is not a prime number. Therefore, the value of \( n = 4 \) disproves the statement.

Would you like further details or have any questions?

### Related Questions:
1. What is a prime number, and how do we identify it?
2. Can you prove that \( n^2 + n + 1 \) is prime for other small values of \( n \)?
3. What is the general method to find when quadratic expressions are prime?
4. Are there any other patterns in numbers that lead to prime results?
5. How can you factor \( n^2 + n + 1 \) for specific values of \( n \)?

### Tip:
When testing mathematical statements, start with small values to check if a counterexample exists quickly.

What value of 'n' will disapprove the statement below? n²+n+1 is a prime number for all n≥1

Explore how to disprove the statement that n² + n + 1 is a prime number for all n≥1 by finding a specific value of n. This solution demonstrates the use of quadratic expressions and prime number identification, suitable for Grades 8-10.

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