[Python] How can I efficiently find integer solutions (x ≠ y) to a Diophantine equation using...

I'm trying to write a Python script to search for integer solutions (x, y) with x ≠ y to the following Diophantine equation:

Here:

• n and k are fixed small positive integers (like n = 1, k = 2),
• x and y range from 1 to a large number (e.g., 1 to 1,000,000),
• I only want solutions where x ≠ y.

Aside from n = 74 and k = 24, for which we get positive integer solutions as desired, how can this code for instance be better written to generate any positive integer solutions to n,k,x,y? Or if possible what other integer solutions exist other than n = 74 and k = 24 ?

I wrote a naive brute-force solution that checks every pair (x, y) in the range, but it's way too slow:

n = 1
k = 2
limit = 10**6

for x in range(1, limit):
rhs = (x + k)**4 - x**4
for y in range(1, limit):
if x == y:
continue
lhs = (y + n)**4 - y**4
if lhs == rhs:
print(f"Match found: x = {x}, y = {y}")

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