{"title":"关于Ahmes级数的几个无理性问题","authors":"V. Kovač, T. Tao","doi":"10.1007/s10474-025-01528-0","DOIUrl":null,"url":null,"abstract":"<div><p> Using basic tools of mathematical analysis and elementary probability\ntheory we address several problems on the irrationality of series of distinct\nunit fractions,<span>\\(\\sum_k 1/a_k\\)</span>. In particular, we study subseries of the Lambert series <span>\\(\\sum_k 1/(t^k-1)\\)</span> and two types of irrationality sequences <span>\\((a_k)\\)</span> introduced by Paul\nErdős and Ronald Graham. Next, we address a question of Erdős, who asked\nhow rapidly a sequence of positive integers <span>\\((a_k)\\)</span> can grow if both series <span>\\(\\sum_k 1/a_k\\)</span> and <span>\\(\\sum_k 1/(a_k+1)\\)</span>have rational sums. Our construction of double exponentially\ngrowing sequences <span>\\((a_k)\\)</span> with this property generalizes to any number <span>\\(d\\)</span> of series<span>\\(\\sum_k 1/(a_k+j)\\)</span>,<span>\\(j=0,1,2,\\ldots,d-1\\)</span>,and, in particular, also gives a positive answer\nto a question of Erdős and Ernst Straus on the interior of the set of <span>\\(d\\)</span>-tuples of their sums.\nFinally, we prove the existence of a sequence <span>\\((a_k)\\)</span> such that all well-defined sums <span>\\(\\sum_k 1/(a_k+t)\\)</span>,<span>\\(t\\in\\mathbb{Z}\\)</span>, are rational numbers, giving a negative answer to a conjecture by Kenneth Stolarsky.</p></div>","PeriodicalId":50894,"journal":{"name":"Acta Mathematica Hungarica","volume":"175 2","pages":"572 - 608"},"PeriodicalIF":0.6000,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On several irrationality problems for Ahmes series\",\"authors\":\"V. Kovač, T. Tao\",\"doi\":\"10.1007/s10474-025-01528-0\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p> Using basic tools of mathematical analysis and elementary probability\\ntheory we address several problems on the irrationality of series of distinct\\nunit fractions,<span>\\\\(\\\\sum_k 1/a_k\\\\)</span>. In particular, we study subseries of the Lambert series <span>\\\\(\\\\sum_k 1/(t^k-1)\\\\)</span> and two types of irrationality sequences <span>\\\\((a_k)\\\\)</span> introduced by Paul\\nErdős and Ronald Graham. Next, we address a question of Erdős, who asked\\nhow rapidly a sequence of positive integers <span>\\\\((a_k)\\\\)</span> can grow if both series <span>\\\\(\\\\sum_k 1/a_k\\\\)</span> and <span>\\\\(\\\\sum_k 1/(a_k+1)\\\\)</span>have rational sums. Our construction of double exponentially\\ngrowing sequences <span>\\\\((a_k)\\\\)</span> with this property generalizes to any number <span>\\\\(d\\\\)</span> of series<span>\\\\(\\\\sum_k 1/(a_k+j)\\\\)</span>,<span>\\\\(j=0,1,2,\\\\ldots,d-1\\\\)</span>,and, in particular, also gives a positive answer\\nto a question of Erdős and Ernst Straus on the interior of the set of <span>\\\\(d\\\\)</span>-tuples of their sums.\\nFinally, we prove the existence of a sequence <span>\\\\((a_k)\\\\)</span> such that all well-defined sums <span>\\\\(\\\\sum_k 1/(a_k+t)\\\\)</span>,<span>\\\\(t\\\\in\\\\mathbb{Z}\\\\)</span>, are rational numbers, giving a negative answer to a conjecture by Kenneth Stolarsky.</p></div>\",\"PeriodicalId\":50894,\"journal\":{\"name\":\"Acta Mathematica Hungarica\",\"volume\":\"175 2\",\"pages\":\"572 - 608\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2025-05-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematica Hungarica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10474-025-01528-0\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Hungarica","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10474-025-01528-0","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
On several irrationality problems for Ahmes series
Using basic tools of mathematical analysis and elementary probability
theory we address several problems on the irrationality of series of distinct
unit fractions,\(\sum_k 1/a_k\). In particular, we study subseries of the Lambert series \(\sum_k 1/(t^k-1)\) and two types of irrationality sequences \((a_k)\) introduced by Paul
Erdős and Ronald Graham. Next, we address a question of Erdős, who asked
how rapidly a sequence of positive integers \((a_k)\) can grow if both series \(\sum_k 1/a_k\) and \(\sum_k 1/(a_k+1)\)have rational sums. Our construction of double exponentially
growing sequences \((a_k)\) with this property generalizes to any number \(d\) of series\(\sum_k 1/(a_k+j)\),\(j=0,1,2,\ldots,d-1\),and, in particular, also gives a positive answer
to a question of Erdős and Ernst Straus on the interior of the set of \(d\)-tuples of their sums.
Finally, we prove the existence of a sequence \((a_k)\) such that all well-defined sums \(\sum_k 1/(a_k+t)\),\(t\in\mathbb{Z}\), are rational numbers, giving a negative answer to a conjecture by Kenneth Stolarsky.
期刊介绍:
Acta Mathematica Hungarica is devoted to publishing research articles of top quality in all areas of pure and applied mathematics as well as in theoretical computer science. The journal is published yearly in three volumes (two issues per volume, in total 6 issues) in both print and electronic formats. Acta Mathematica Hungarica (formerly Acta Mathematica Academiae Scientiarum Hungaricae) was founded in 1950 by the Hungarian Academy of Sciences.