A field is made of $2017 \times 2017$ unit squares. Luffy has $k$ gold detectors, which he places on some of the unit squares, then he leaves the area. Sanji then chooses a $1500 \times 1500$ area, then buries a gold coin on each unit square in this area and none other. When Luffy returns, a gold detector beeps if and only if there is a gold coin buried underneath the unit square it's on. It turns out that by an appropriate placement, Luffy will always be able to determine the $1500 \times 1500$ area containing the gold coins by observing the detectors, no matter how Sanji places the gold coins. Determine the minimum value of $k$ in which this is possible.
1. **Define the problem and setup:**
   - We have a $2017 \times 2017$ grid of unit squares.
   - Luffy has $k$ gold detectors.
   - Sanji chooses a $1500 \times 1500$ area to bury gold coins.
   - Luffy needs to determine the $1500 \times 1500$ area using the detectors.

2. **Objective:**
   - Determine the minimum value of $k$ such that Luffy can always determine the $1500 \times 1500$ area.

3. **Coordinate System:**
   - Represent the coordinates of the squares with elements of the set $\{1, 2, \dots, 2017\} \times \{1, 2, \dots, 2017\}$.

4. **Detector Placement Strategy:**
   - Consider placing detectors at the points $\{(1, 518), (2, 518), \dots, (517, 518)\}$ and $\{(518, 1), (518, 2), \dots, (518, 517)\}$.
   - This placement ensures that any $1500 \times 1500$ area will intersect with at least one detector.

5. **Partition the Grid:**
   - Partition the grid into 9 rectangles using the sets of coordinates $A_{i,j} = S_i \times S_j$ where $1 \leq i, j \leq 3$.
   - Define $S_1 = \{1, 2, \dots, 517\}$, $S_2 = \{518, 519, \dots, 1500\}$, $S_3 = \{1501, 1502, \dots, 2017\}$.

6. **Inequalities for Detector Counts:**
   - Let $K(A_{i,j})$ be the number of gold detectors in the rectangle $A_{i,j}$.
   - From the observation, we have:
     \[
     K(A_{i,j}) + K(A_{i,j+1}) + K(A_{i+2,j}) + K(A_{i+2,j+1}) \geq 517
     \]
     \[
     K(A_{i,j}) + K(A_{i+1,j}) + K(A_{i,j+2}) + K(A_{i+1,j+2}) \geq 517
     \]

7. **Summing the Inequalities:**
   - Adding up all the inequalities produced:
     \[
     2(-K(A_{2,2}) + \sum_{1 \leq i, j \leq 3} K(A_{i,j})) \geq 4 \cdot 517
     \]
   - Simplifying, we get:
     \[
     \sum_{1 \leq i, j \leq 3} K(A_{i,j}) \geq 1034
     \]

8. **Conclusion:**
   - The minimum value of $k$ such that Luffy can always determine the $1500 \times 1500$ area is $1034$.

The final answer is $\boxed{1034}$