Answer
See the explanation
Work Step by Step
To find integer solutions for equations of the form \( x^{2}+y^{2}=n \), where \( n \) is a given positive integer, you can use the following algorithm:
1. Initialize an empty list to store the solutions.
2. Iterate through all possible values of \( x \) from 0 to \(\sqrt{n}\).
3. For each value of \( x \), calculate \( y = \sqrt{n - x^{2}} \).
4. If \( y \) is an integer, add the pair \( (x, y) \) to the list of solutions.
Here's a Python code snippet implementing this algorithm:
```python
import math
def find_integer_solutions(n):
solutions = []
for x in range(int(math.sqrt(n)) + 1):
y_squared = n - x**2
y = int(math.sqrt(y_squared))
if y**2 == y_squared:
solutions.append((x, y))
return solutions
# Example usage:
n = 25
print(find_integer_solutions(n))
```
The time complexity of this algorithm can be analyzed as follows:
- The loop runs from 0 to \(\sqrt{n}\), which has a time complexity of \( O(\sqrt{n}) \).
- Inside the loop, calculating \( y \) and checking if it's an integer can be done in constant time.
- Therefore, the overall time complexity of the algorithm is \( O(\sqrt{n}) \).
