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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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  • $\begingroup$ I guess you mean $\sum_{i=1}^{t}p_{i}^{k}=N$ ? $\endgroup$
    – Sylvain JULIEN
    Commented Aug 12, 2011 at 10:19
  • $\begingroup$ Yes you are correct! $\endgroup$
    – user16007
    Commented Aug 12, 2011 at 13:13
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    $\begingroup$ 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. $\endgroup$
    – Péter Komjáth
    Commented Aug 13, 2011 at 7:20

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