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I'm looking for information on this kind of reverse-Fermat problem: Given $a,b,c \in \mathbb{C}$, find a $z \in \mathbb{C}$ such that $a^z + b^z = c^z$?

When does such a $z$ exist? Is anything known? Thanks.

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    Sure, but it takes on$a$discrete set of values, and it's still reasonable to ask if we can find such a $z$ so as to make the equality hold for an appropriate choice of branches.2011-09-27

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