Sure, I can write a test case generator for the given problem statement. The generator will create random values for \(N\), \(x\), and the list \(a\) according to the constraints specified in the problem.

Here is the test case generator:

```python
import random

def tcgen():
    # Generate a random value for N within the range [2, 100]
    N = random.randint(2, 100)
    
    # Generate a random value for x within the range [1, 10^9]
    x = random.randint(1, 10**9)
    
    # Generate a list a of size N, with each element within the range [1, 10^9]
    a = [random.randint(1, 10**9) for _ in range(N)]
    
    return N, x, a

# Generating a random test case
test_case = tcgen()
print("N:", test_case[0])
print("x:", test_case[1])
print("a:", test_case[2])
```

This function will generate random test cases that adhere to the constraints:

- \(N\) is an integer between 2 and 100.
- \(x\) is an integer between 1 and \(10^9\).
- Each \(a_i\) is an integer between 1 and \(10^9\).

You can run the `tcgen()` function to generate a new test case each time. The output will be in the format required by the problem statement, with \(N\), \(x\), and the list \(a\).