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Given a number $N$ is it possible to determine that the number can be splitted as a sum of $c$ perfect powers i.e.


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Can the powers be different or do they have to be the same? If same then the general results are any number is sum of 4 squares, 9 cubes, 19 fourth powers etc. This is called the Waring Problem. –  user44197 Feb 4 at 4:45
The powers can be different –  Tom Lynd Feb 4 at 6:32
Are the $a_i$ non-negative? If so, the exponents are bounded by $\log N / \log 2$, and so there are only finitely many possibilities to check, making it very possible to determine what you ask. –  Matthew Conroy Feb 4 at 20:06

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