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It's well-known that every natural number can be written as a sum of 4 squares of integers.

Has there been any recent progress about the similar problem for the cubes, 4-th powers and so on? I believe this was proven to be representable using some N that depends on the power and what's the story about it?

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Sums of cubes and more

It's well-known that every natural number can be written as a sum of 4 squares of integers.

Has there been any recent progress about the similar problem for the cubes, 4-th powers and so on?