{"title":"Sums of One Prime Power and Four Prime Cubes in Short Intervals","authors":"Gen Li, Xianjiu Huang, Xiaoming Pan, Li Yang","doi":"10.1155/2023/3244257","DOIUrl":null,"url":null,"abstract":"<jats:p>Let <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\n <mi>k</mi>\n <mo>⩾</mo>\n <mn>1</mn>\n </math>\n </jats:inline-formula> be an integer. In this study, we derive an asymptotic formula for the average number of representations of integers <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\">\n <mi>n</mi>\n <mo>=</mo>\n <msubsup>\n <mrow>\n <mi>p</mi>\n </mrow>\n <mrow>\n <mn>1</mn>\n </mrow>\n <mrow>\n <mi>k</mi>\n </mrow>\n </msubsup>\n <mo>+</mo>\n <msubsup>\n <mrow>\n <mi>p</mi>\n </mrow>\n <mrow>\n <mn>2</mn>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </msubsup>\n <mo>+</mo>\n <msubsup>\n <mrow>\n <mi>p</mi>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </msubsup>\n <mo>+</mo>\n <msubsup>\n <mrow>\n <mi>p</mi>\n </mrow>\n <mrow>\n <mn>4</mn>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </msubsup>\n <mo>+</mo>\n <msubsup>\n <mrow>\n <mi>p</mi>\n </mrow>\n <mrow>\n <mn>5</mn>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </msubsup>\n </math>\n </jats:inline-formula> in short intervals, where <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\">\n <msub>\n <mrow>\n <mi>p</mi>\n </mrow>\n <mrow>\n <mn>1</mn>\n </mrow>\n </msub>\n <mo>,</mo>\n <msub>\n <mrow>\n <mi>p</mi>\n </mrow>\n <mrow>\n <mn>2</mn>\n </mrow>\n </msub>\n <mo>,</mo>\n <msub>\n <mrow>\n <mi>p</mi>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </msub>\n <mo>,</mo>\n <msub>\n <mrow>\n <mi>p</mi>\n </mrow>\n <mrow>\n <mn>4</mn>\n </mrow>\n </msub>\n <mo>,</mo>\n <msub>\n <mrow>\n <mi>p</mi>\n </mrow>\n <mrow>\n <mn>5</mn>\n </mrow>\n </msub>\n </math>\n </jats:inline-formula> are prime numbers.</jats:p>","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Muenster Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1155/2023/3244257","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let be an integer. In this study, we derive an asymptotic formula for the average number of representations of integers in short intervals, where are prime numbers.
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