The equation a^n + b^n = c^n easily has lots solutions with real numbers. With monumental effort, Wiles et al. showed it has no solutions with integers n>2, and a trivial corollary is that it has no solutions with rational numbers either.
It easily has lots of solutions with algebraic numbers (just take the nth root). Where is the boundary, the threshold size of the subset of reals, where it changes from having lots of solutions to few or none? Constructible numbers, quadratic surds?
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