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<p>Estimates are obtained for the number of natural numbers <i>n</i> in certain residue classes that do not have a representation of the form <Equation ID="Equ1"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11464_2024_52_Article_Equ1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="124" /> </MediaObject> <EquationSource Format="TEX">\(n = {p_1} + p_2^2 + p_3^k,\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>n</mi> <mo>=</mo> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> </mrow> <mo>+</mo> <msubsup> <mi>p</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>p</mi> <mn>3</mn> <mi>k</mi> </msubsup> <mo>,</mo> </math></EquationSource> </Equation> for <i>k</i> = 3 or <i>k</i> = 4 respectively, where <i>p</i><sub><i>i</i></sub> are primes. We improve the previous results in [Monatsh. Math., 2019, 188(2): 269–285].</p>

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On Exceptional Sets for Waring-Goldbach Problems with Unlike Powers

  • 期刊论文

  • 发表日期：

    2025年1月5日

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  • Rui Zhang,
  • Wei Zhang

摘要

Estimates are obtained for the number of natural numbers n in certain residue classes that do not have a representation of the form \(n = {p_1} + p_2^2 + p_3^k,\) n = p 1 + p 2 2 + p 3 k , for k = 3 or k = 4 respectively, where pi are primes. We improve the previous results in [Monatsh. Math., 2019, 188(2): 269–285].