[BarkingMad]

I'm no mathmetician. So, maybe some of the criteria flew over my head, but I plugged the formula into Excel, inserted some simple numbers for the A-B-C values, and used solver to find "Q".

For N = 5
A = 5
B = 4
C = 4
Q = 3125

5^5 + 4^5 = 4149

4^5 + 3125 = 4149

Again, maybe some of the constraints of the "conjecture" flew over my head. If I understood it correctly though, it seems like there are many Q's for which the conjecture would hold true.