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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
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Lower bounds on the easier Waring problem
As far as I am aware, nothing unconditional is known. The difficulty is that one has no obvious constraint on the size of the variables, so that the $x_i$ could be arbitrarily large in terms of $n$ in …