Mike Bennett, Nathan Ng, and I posted a paper recently about solutions to the equation

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:

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