Question

Let the set \( C = \{(x, y) \mid x^2 - 2^y = 2023, x, y \in \mathbb{N}\} \).Then\[\sum_{(x, y) \in C} (x + y)\]is equal to ______.

Answer 1

Correct Answer: 46

Considering the equation: \(x^2 - 2^y = 2023\)

Step 1. Verification through substitution indicates that \( x = 45 \) and \( y = 1 \) satisfy the equation, evidenced by:  

  \(45^2 - 2^1 = 2025 - 2 = 2023\)
 

Step 2. Consequently, the unique solution within the set \( C \) is identified as \( (x, y) = (45, 1) \).

Step 3. The summation of \( (x + y) \) for all pairs \( (x, y) \) in \( C \) is calculated as:  

  \(\sum_{(x, y) \in C} (x + y) = 45 + 1 = 46\)
 

The Correct Answer is: 46