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Current status of Waring-Goldbach problem

http://en.wikipedia.org/wiki/Waring%E2%80%93Goldbach_problem

Wiki says that WG conjecture is that for every $k$, $\exists$ primes $p_{1}, p_{2}, \cdots p_{t}$ where $t$ is independent of $k$ such that $\sum_{i=1}^{t}p_{i}^{k} = N$ for large enough $N$.

Hua showed $t$ is atmost $O(k^{2}log{k})$. In his theorem, does anyone know the bounds on $N$ given $k$? Is it bounded below $O(2^{k})$ (that is for every $N > N_{0} = O(2^{k})$ does Hua's theorem hold)?

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I guess you mean $\sum_{i=1}^{t}p_{i}^{k}=N$ ? –  Sylvain JULIEN Aug 12 '11 at 10:19
    
Yes you are correct! –  user16007 Aug 12 '11 at 13:13
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I am not familiar with Hua's or other people's proofs, but they may well be ineffective, as they most probably use the Siegel-Walfisz theorem, which is ineffective. This was the case with Vinogradov's original proof of the odd Goldbach conjecture. –  Péter Komjáth Aug 13 '11 at 7:20

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