Let the set `C = {(x, y) | x^2- 2^y = 2023`, `x, y in }`. Then `sum_((x,y) in C) (x + y)` is equal to _______.

Answer

To solve the problem, we need to find the set \( C = \{(x, y) | x^2 - 2^y = 2023, x, y \in \mathbb{Z}\} \) and then compute the sum \( \sum_{(x,y) \in C} (x + y) \).

### Step-by-step Solution:

1. **Rearranging the Equation**:
We start with the equation given in the problem:
\[ x^2 - 2^y = 2023 \]
Rearranging gives us:
\[ x^2 = 2023 + 2^y \]

2. **Finding Possible Values for \( x^2 \)**:
Since \( x^2 \) must be a perfect square, \( 2023 + 2^y \) must also be a perfect square. We will check various values of \( y \) to find corresponding values of \( x \).

3. **Testing Values of \( y \)**:
We will try different integer values for \( y \) to see if \( 2023 + 2^y \) results in a perfect square.

- For \( y = 0 \):
\[ x^2 = 2023 + 2^0 = 2024 \quad (\text{not a perfect square}) \]
- For \( y = 1 \):
\[ x^2 = 2023 + 2^1 = 2025 \quad (x = 45) \]
- For \( y = 2 \):
\[ x^2 = 2023 + 2^2 = 2027 \quad (\text{not a perfect square}) \]
- For \( y = 3 \):
\[ x^2 = 2023 + 2^3 = 2031 \quad (\text{not a perfect square}) \]
- For \( y = 4 \):
\[ x^2 = 2023 + 2^4 = 2039 \quad (\text{not a perfect square}) \]
- For \( y = 5 \):
\[ x^2 = 2023 + 2^5 = 2051 \quad (\text{not a perfect square}) \]
- For \( y = 6 \):
\[ x^2 = 2023 + 2^6 = 2055 \quad (\text{not a perfect square}) \]
- For \( y = 7 \):
\[ x^2 = 2023 + 2^7 = 2071 \quad (\text{not a perfect square}) \]
- For \( y = 8 \):
\[ x^2 = 2023 + 2^8 = 2087 \quad (\text{not a perfect square}) \]
- For \( y = 9 \):
\[ x^2 = 2023 + 2^9 = 2111 \quad (\text{not a perfect square}) \]
- For \( y = 10 \):
\[ x^2 = 2023 + 2^{10} = 2039 \quad (\text{not a perfect square}) \]

After testing these values, we find that the only valid pair is when \( y = 1 \) and \( x = 45 \).

4. **Calculating the Sum**:
Now, we calculate \( x + y \):
\[ x + y = 45 + 1 = 46 \]

5. **Conclusion**:
Since we only found one valid pair \( (45, 1) \), the final answer is:
\[ \sum_{(x,y) \in C} (x + y) = 46 \]

### Final Answer:
46