Given a fixed positive integer k > 1, find all integer solutions to the equation y^k = x^2 + x.
(x^y means x to the power of y)
WHY NO ANSWERS
To find all integer solutions to the equation y^k = x^2 + x, we can try different values of x and y and check if the equation is satisfied. Let's use a brute-force approach to find the solutions.
1. Start by initializing two variables, x and y, to 0.
2. Iterate x from 0 to infinity.
3. For each value of x, iterate y from 0 to infinity.
4. Calculate the right side of the equation, x^2 + x.
5. Calculate y^k.
6. If y^k is equal to x^2 + x, we have found a solution. Print the values of x and y.
7. Repeat steps 4-6 until you have exhausted all possible values of x and y.
Here's a Python code snippet that implements this approach:
```python
k = int(input("Enter a positive integer k: "))
x = 0
while True:
y = 0
while True:
right_side = x**2 + x
left_side = y**k
if left_side > right_side:
break
elif left_side == right_side:
print("Solution found: x =", x, "y =", y)
y += 1
x += 1
```
Keep in mind that this brute-force method may take a long time to find solutions for large values of k. Consider optimizing the algorithm if you need to find solutions for large values of k.