{"title":"Unit fractions with shifted prime denominators","authors":"Thomas F. Bloom","doi":"10.1017/prm.2024.42","DOIUrl":null,"url":null,"abstract":"<p>We prove that any positive rational number is the sum of distinct unit fractions with denominators in <span><span><span data-mathjax-type=\"texmath\"><span>$\\{p-1 : p\\textrm { prime}\\}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240401120835823-0350:S0308210524000428:S0308210524000428_inline1.png\"/></span></span>. The same conclusion holds for the set <span><span><span data-mathjax-type=\"texmath\"><span>$\\{p-h : p\\textrm { prime}\\}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240401120835823-0350:S0308210524000428:S0308210524000428_inline2.png\"/></span></span> for any <span><span><span data-mathjax-type=\"texmath\"><span>$h\\in \\mathbb {Z}\\backslash \\{0\\}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240401120835823-0350:S0308210524000428:S0308210524000428_inline3.png\"/></span></span>, provided a necessary congruence condition is satisfied. We also prove that this is true for any subset of the primes of relative positive density, provided a necessary congruence condition is satisfied.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"47 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/prm.2024.42","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We prove that any positive rational number is the sum of distinct unit fractions with denominators in $\{p-1 : p\textrm { prime}\}$. The same conclusion holds for the set $\{p-h : p\textrm { prime}\}$ for any $h\in \mathbb {Z}\backslash \{0\}$, provided a necessary congruence condition is satisfied. We also prove that this is true for any subset of the primes of relative positive density, provided a necessary congruence condition is satisfied.
期刊介绍：
A flagship publication of The Royal Society of Edinburgh, Proceedings A is a prestigious, general mathematics journal publishing peer-reviewed papers of international standard across the whole spectrum of mathematics, but with the emphasis on applied analysis and differential equations.
An international journal, publishing six issues per year, Proceedings A has been publishing the highest-quality mathematical research since 1884. Recent issues have included a wealth of key contributors and considered research papers.
. The same conclusion holds for the set $\{p-h : p\textrm { prime}\}$
for any $h\in \mathbb {Z}\backslash \{0\}$
, provided a necessary congruence condition is satisfied. We also prove that this is true for any subset of the primes of relative positive density, provided a necessary congruence condition is satisfied.
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