I'm surprised that Bjorn Poonen hasn't chimed in yet. However, you can read about the approaches to solving equations like this in Frits Beuker's lectures "The generalized Fermat Equation" http://www.math.uu.nl/people/beukers/Fermatlectures.pdf
Added later: You should also look at Siksek's "Edingurgh lectures" for a good outline of how he might have approached this equation: http://www.warwick.ac.uk/~maseap/papers/edinburgh3.pdf