To identify 
𝑛
n values that produce prime numbers using the formula 
𝑥
=
90
𝑛
+
47
x=90n+47, follow these steps:

Avoid Multiples of Certain Numbers:

Do not use 
𝑛
n values that are multiples of 4, 5, or 7. This means if 
𝑛
n can be evenly divided by 4, 5, or 7, skip it. Check Residues (Remainders):

When 
𝑛
n is divided by 4, it should not leave a remainder of 0 or 2.
When 
𝑛
n is divided by 5, it should not leave a remainder of 0 or 2.
When 
𝑛
n is divided by 7, it should not leave a remainder of 0, 2, 4, or 5.

To identify \( n \) values that produce prime numbers using the formula \( x = 90n + 47 \), you need to ensure that \( n \) does not meet certain divisibility criteria to avoid non-prime numbers. Here's the detailed process:

### Avoid Multiples of Certain Numbers:
- Do not use \( n \) values that are multiples of 4, 5, or 7.

### Check Residues (Remainders):
- When \( n \) is divided by 4, it should not leave a remainder of 0 or 2.
- When \( n \) is divided by 5, it should not leave a remainder of 0 or 2.
- When \( n \) is divided by 7, it should not leave a remainder of 0, 2, 4, or 5.

### Example Calculation:
Let's verify a few \( n \) values to check if they produce a prime number.

1. **For \( n = 1 \):**
   - \( x = 90 \cdot 1 + 47 = 137 \)
   - Check if 137 is a prime number.
   - 137 is a prime number.

2. **For \( n = 2 \):**
   - \( x = 90 \cdot 2 + 47 = 227 \)
   - Check if 227 is a prime number.
   - 227 is a prime number.

3. **For \( n = 3 \):**
   - \( x = 90 \cdot 3 + 47 = 317 \)
   - Check if 317 is a prime number.
   - 317 is a prime number.

4. **For \( n = 4 \):**
   - \( x = 90 \cdot 4 + 47 = 407 \)
   - Check if 407 is a prime number.
   - 407 is not a prime number (divisible by 11).

5. **For \( n = 5 \):**
   - \( x = 90 \cdot 5 + 47 = 497 \)
   - Check if 497 is a prime number.
   - 497 is not a prime number (divisible by 7).

### General Algorithm:
1. Iterate over possible \( n \) values.
2. Check if \( n \) is a multiple of 4, 5, or 7.
3. If not, check the remainders when \( n \) is divided by 4, 5, and 7.
4. Calculate \( x \) and check if \( x \) is a prime number.
5. Return the \( n \) values that produce prime \( x \).

### Example Python Code:
Here's a simple Python script to automate this process:

```python
import sympy

def is_valid_n(n):
    if n % 4 == 0 or n % 4 == 2:
        return False
    if n % 5 == 0 or n % 5 == 2:
        return False
    if n % 7 == 0 or n % 7 == 2 or n % 7 == 4 or n % 7 == 5:
        return False
    return True

def find_prime_generating_ns(limit):
    prime_generating_ns = []
    for n in range(1, limit + 1):
        if is_valid_n(n):
            x = 90 * n + 47
            if sympy.isprime(x):
                prime_generating_ns.append(n)
    return prime_generating_ns

# Example usage:
limit = 1000  # Define the range limit
prime_generating_ns = find_prime_generating_ns(limit)
print(prime_generating_ns)
```

This script will output \( n \) values within the given limit that produce prime numbers using the formula \( x = 90n + 47 \).

Would you like more details or have any questions about the process?

Here are 5 questions you might consider asking next:
1. How can I modify the script to handle larger ranges more efficiently?
2. What are some common prime-checking algorithms and their efficiencies?
3. How can I visualize the distribution of \( n \) values that produce prime numbers?
4. Can you provide a proof or reasoning for the residue conditions?
5. How do prime numbers generated by this formula compare to random prime numbers?

**Tip:** When testing for prime numbers in large datasets, consider using probabilistic methods like the Miller-Rabin test for efficiency.

Discover how to find prime numbers using the formula x = 90n + 47. Learn about avoiding multiples of 4, 5, and 7, and checking residues for n. Suitable for advanced math enthusiasts interested in number theory and modular arithmetic.

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