Let ##IMG## [http://ej.iop.org/images/1064-5616/211/5/733/MSB_211_5_733ieqn1.gif] {$\mathbb{N}_{0}$} be the set of positive integers whose binary decompositions contain an even number of ones. We give a bound for the trigonometric sum of special form over numbers in ##IMG## [http://ej.iop.org/images/1064-5616/211/5/733/MSB_211_5_733ieqn1.gif] {$\mathbb{N}_{0}$} ; using this bound, we derive an asymptotic formula for the number of solutions to Waring’s equation in positive integers in ##IMG## [http://ej.iop.org/images/1064-5616/211/5/733/MSB_211_5_733ieqn1.gif] {$\mathbb{N}_{0}$} , and also a bound for the number of terms in the last equation, which is sufficient for the equation to be solvable in integers in ##IMG## [http://ej.iop.org/images/1064-5616/211/5/733/MSB_211_5_733ieqn1.gif] {$\mathbb{N}_{0}$} . Bibliography: 9 titles.

中文翻译：

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特殊形式的自然数的沃林问题

让## IMG ## [http://ej.iop.org/images/1064-5616/211/5/733/MSB_211_5_733ieqn1.gif] {$ \ mathbb {N} _ {0} $}为正整数，其二进制分解包含一个偶数。我们在## IMG ## [http://ej.iop.org/images/1064-5616/211/5/733/MSB_211_5_733ieqn1.gif] {$ \ mathbb {N} _ {0} $}; 使用此界限，我们在## IMG ## [http://ej.iop.org/images/1064-5616/211/5/733/MSB_211_5_733ieqn1中以正整数形式导出Waring方程的解数的渐近公式.gif] {$ \ mathbb {N} _ {0} $}，也限制了最后一个方程式中的项数，这足以使该方程式可以在## IMG ## [http ：//ej.iop.org/images/1064-5616/211/5/733/MSB_211_5_733ieqn1.gif] {$ \ mathbb {N} _ {0} $}。参考书目：9种。