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On Hilbert’s solution of Waring’s problem

100%
Pollack P.
Open Mathematics
|
2011
|
tom 9
|
nr 2
294-301
EN
In 1909, Hilbert proved that for each fixed k, there is a number g with the following property: Every integer N ≥ 0 has a representation in the form N = x 1k + x 2k + … + x gk, where the x i are nonnegative integers. This resolved a conjecture of Edward Waring from 1770. Hilbert’s proof is somewhat unsatisfying, in that no method is given for finding a value of g corresponding to a given k. In his doctoral thesis, Rieger showed that by a suitable modification of Hilbert’s proof, one can give explicit bounds on the least permissible value of g. We show how to modify Rieger’s argument, using ideas of F. Dress, to obtain a better explicit bound. While far stronger bounds are available from the powerful Hardy-Littlewood circle method, it seems of some methodological interest to examine how far elementary techniques of this nature can be pushed.
2
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On pairs of equations in unlike powers of primes and powers of 2

61%
Hu L., Yang L.
Open Mathematics
|
2017
|
tom 15
|
nr 1
1487-1494
EN
In this paper, we obtained that when k = 455, every pair of large even integers satisfying some necessary conditions can be represented in the form of a pair of unlike powers of primes and k powers of 2.
3
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Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem

52%
Feng Z.
Open Mathematics
|
2017
|
tom 15
|
nr 1
1517-1529
EN
In this paper, we are able to prove that almost all integers n satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 7, 8, i.e., [...] N=p13+…+pj3 $\begin{array}{} N=p_1^3+ \ldots +p_j^3 \end{array} $ with [...] |pi−(N/j)1/3|≤N1/3−δ+ε(1≤i≤j), $\begin{array}{} |p_i-(N/j)^{1/3}|\leq N^{1/3- \delta +\varepsilon} (1\leq i\leq j), \end{array} $ for some [...] 0<δ≤190. $\begin{array}{} 0 \lt \delta\leq\frac{1}{90}. \end{array} $ Furthermore, we give the quantitative relations between the length of short intervals and the size of exceptional sets.
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