A^4 + B^2 = C^n (*)
in relatively prime integers (A,B,C). There are infinitely many solutions to this equation when n = 2 or 3; for example,
1089^4 + 1549034^2 = 15613^3.
By contrast, what Mike, Nathan and I prove is that for n > 3, there are no solutions to (*).
(CORRECTION: Mrs. Q points out that I need to say “relatively prime nonzero integers” above — otherwise there are solutions like 0^4 + 1^2 = 1^n.)
More math below the fold: