I need to find integer solutions to the equation a^a + b^b = c^c. I could resort to traditional programming in, say, the Python programming language, where I would use three **for** loops (followed by some basic mathematical operations) and check to see whether the sum on the left of the equation is exactly equal to the value on the right. Assuming I iterate each of the three loops from 0 to 1000, the program will need to check through a little over 1 billion possible integer combinations (calculated by 1001^3). Due to the symmetry of a^a + b^b, a better optimization would be to only work with scenarios where a<=b, in which case the program will need to check through a little over 502 million possible integer combinations. I can optimize further. Take the sum on the left and attempt to resolve it to c^c. Pythonâ€™s gmpy2 library will easily do that for us; just iterate the exponent from **2** to **log base 2 of the sum** (itâ€™s a trick from mathematics). See this example:

```
import math
import gmpy2
gmpy2.get_context().precision=1000 #The higher the better, but slower
sum = 285311670611 #Will be obtained from a^a + b^b. This is just an example
for c in range(2, round(math.log(sum, 2))+2):
result = gmpy2.root(sum, c) #This is where the magic happens!
if(result%1==0 and result==c):
print(result)
```

This is great and all but the number of integer combinations increases as the range of a,b increases. So, should I (can I) resort to GPU programming to try and quickly find solutions to the equation? If you have any questions, ask away and I will gladly answer them. Thank you.