Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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GREEN'S ADDITIVE COMPLEMENT PROBLEM FOR k-TH POWERS

  • Ding, Yuchen (School of Mathematical Science Yangzhou University) ;
  • Wang, Li-Yuan (School of Physical and Mathematical Sciences Nanjing Tech University)
  • Received : 2021.02.20
  • Accepted : 2021.12.31
  • Published : 2022.03.01
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Abstract

Let k ⩾ 2 be an integer, Sk = {1k, 2k, 3k, …} and B = {b1, b2, b3, …} be an additive complement of Sk, which means all sufficiently large integers can be written as the sum of an element of Sk and an element of B. In this paper we prove that $${{\lim}\;{\sup}}\limits_{n{\rightarrow}{\infty}}\;{\frac{{\Gamma}(2-{\frac{1}{k}})^{\frac{k}{k-1}}{\Gamma}(1+{\frac{1}{k}})^{\frac{k}{k-1}}n^{\frac{k}{k-1}}-b_n}{n}}\;{\geqslant}\;{\frac{k}{2(k-1)}}\;{\frac{{\Gamma}(2-{\frac{1}{k}})^2}{{\Gamma}(2-{\frac{2}{k}})}},$$ where 𝚪(·) is Euler's Gamma function.

Keywords

Acknowledgement

The first author was supported by the Natural Science Foundation of Jiangsu Province of China (No. BK20210784). He was also supported by the foundations of the projects "Jiangsu Provincial Double-Innovation Doctor Program" (No. JSSCBS20211023) and "Golden Phenix of the Green City-Yang Zhou" to excellent PhD (No. YZLYJF2020PHD051). The second author was supported by the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (Grant No. 21KJB110001).

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