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Given a number $N$, is it possible to determine whether the number can be splitted to a sum of $c$ perfect powers:


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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 '14 at 4:45
The powers can be different – Tom Lynd Feb 4 '14 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 '14 at 20:06
Trivial even for c = 1 if an exponent of 1 is allowed :-) – gnasher729 Apr 13 at 15:09

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