类collatz函数的一个新的用素数条件证明有理循环

IF 0.7 Q2 MATHEMATICS
Benjamin Bairrington, Nabil Mohsen
{"title":"类collatz函数的一个新的用素数条件证明有理循环","authors":"Benjamin Bairrington, Nabil Mohsen","doi":"10.1155/2023/5159528","DOIUrl":null,"url":null,"abstract":"<jats:p>In this paper, we study the bounded trajectories of Collatz-like functions. Fix <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\n <mi>α</mi>\n <mo>,</mo>\n <mi>β</mi>\n <mo>∈</mo>\n <msub>\n <mrow>\n <mi mathvariant=\"double-struck\">Z</mi>\n </mrow>\n <mrow>\n <mo>></mo>\n <mn>0</mn>\n </mrow>\n </msub>\n </math>\n </jats:inline-formula> so that <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\">\n <mi>α</mi>\n </math>\n </jats:inline-formula> and <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\">\n <mi>β</mi>\n </math>\n </jats:inline-formula> are coprime. Let <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\">\n <mover accent=\"true\">\n <mi>k</mi>\n <mo>¯</mo>\n </mover>\n <mo>=</mo>\n <mfenced open=\"(\" close=\")\" separators=\"|\">\n <mrow>\n <msub>\n <mrow>\n <mi>k</mi>\n </mrow>\n <mrow>\n <mn>1</mn>\n </mrow>\n </msub>\n <mo>,</mo>\n <mo>…</mo>\n <mo>,</mo>\n <msub>\n <mrow>\n <mi>k</mi>\n </mrow>\n <mrow>\n <mi>β</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msub>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula> so that for each <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M5\">\n <mn>1</mn>\n <mo>≤</mo>\n <mi>i</mi>\n <mo>≤</mo>\n <mi>β</mi>\n <mo>−</mo>\n <mn>1</mn>\n </math>\n </jats:inline-formula>, <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M6\">\n <msub>\n <mrow>\n <mi>k</mi>\n </mrow>\n <mrow>\n <mi>i</mi>\n </mrow>\n </msub>\n <mo>∈</mo>\n <msub>\n <mrow>\n <mi mathvariant=\"double-struck\">Z</mi>\n </mrow>\n <mrow>\n <mo>></mo>\n <mn>0</mn>\n </mrow>\n </msub>\n </math>\n </jats:inline-formula>, <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M7\">\n <msub>\n <mrow>\n <mi>k</mi>\n </mrow>\n <mrow>\n <mi>i</mi>\n </mrow>\n </msub>\n </math>\n </jats:inline-formula> is coprime to <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M8\">\n <mi>α</mi>\n </math>\n </jats:inline-formula> and <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M9\">\n <mi>β</mi>\n </math>\n </jats:inline-formula>, and <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M10\">\n <msub>\n <mrow>\n <mi>k</mi>\n </mrow>\n <mrow>\n <mi>i</mi>\n </mrow>\n </msub>\n <mo>≡</mo>\n <mi>i</mi>\n <mtext> </mtext>\n <mfenced open=\"(\" close=\")\" separators=\"|\">\n <mrow>\n <mi mathvariant=\"normal\">mod</mi>\n <mtext> </mtext>\n <mi>β</mi>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula>. We define the function <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M11\">\n <msub>\n <mrow>\n <mi>C</mi>\n </mrow>\n <mrow>\n <mfenced open=\"(\" close=\")\" separators=\"|\">\n <mrow>\n <mi>α</mi>\n <mo>,</mo>\n <mi>β</mi>\n <mo>,</mo>\n <mover accent=\"true\">\n <mi>k</mi>\n <mo>¯</mo>\n </mover>\n </mrow>\n </mfenced>\n </mrow>\n </msub>\n <mo>:</mo>\n <msub>\n <mrow>\n <mi mathvariant=\"double-struck\">Z</mi>\n </mrow>\n <mrow>\n <mo>></mo>\n <mn>0</mn>\n </mrow>\n </msub>\n <mo>⟶</mo>\n <msub>\n <mrow>\n <mi mathvariant=\"double-struck\">Z</mi>\n </mrow>\n <mrow>\n <mo>></mo>\n <mn>0</mn>\n </mrow>\n </msub>\n </math>\n </jats:inline-formula> and the sequence <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M12\">\n <mfenced open=\"{\" close=\"}\" separators=\"|\">\n <mrow>\n <mi>n</mi>\n <mo>,</mo>\n <msub>\n <mrow>\n <mi>C</mi>\n </mrow>\n <mrow>\n <mfenced open=\"(\" close=\")\" separators=\"|\">\n <mrow>\n <mi>α</mi>\n <mo>,</mo>\n <mi>β</mi>\n <mo>,</mo>\n <mover accent=\"true\">\n <mi>k</mi>\n <mo>¯</mo>\n </mover>\n </mrow>\n </mfenced>\n </mrow>\n </msub>\n <mrow>\n <mfenced open=\"(\" close=\")\" separators=\"|\">\n <mrow>\n <mi>n</mi>\n </mrow>\n </mfenced>\n </mrow>\n <mo>,</mo>\n <msubsup>\n <mrow>\n <mi>C</mi>\n </mrow>\n <mrow>\n <mfenced open=\"(\" close=\")\" separators=\"|\">\n <mrow>\n <mi>α</mi>\n <mo>,</mo>\n <mi>β</mi>\n <mo>,</mo>\n <mover accent=\"true\">\n <mi>k</mi>\n <mo>¯</mo>\n </mover>\n </mrow>\n </mfenced>\n </mrow>\n ","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A New Proof of Rational Cycles for Collatz-Like Functions Using a Coprime Condition\",\"authors\":\"Benjamin Bairrington, Nabil Mohsen\",\"doi\":\"10.1155/2023/5159528\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<jats:p>In this paper, we study the bounded trajectories of Collatz-like functions. Fix <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M1\\\">\\n <mi>α</mi>\\n <mo>,</mo>\\n <mi>β</mi>\\n <mo>∈</mo>\\n <msub>\\n <mrow>\\n <mi mathvariant=\\\"double-struck\\\">Z</mi>\\n </mrow>\\n <mrow>\\n <mo>></mo>\\n <mn>0</mn>\\n </mrow>\\n </msub>\\n </math>\\n </jats:inline-formula> so that <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M2\\\">\\n <mi>α</mi>\\n </math>\\n </jats:inline-formula> and <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M3\\\">\\n <mi>β</mi>\\n </math>\\n </jats:inline-formula> are coprime. Let <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M4\\\">\\n <mover accent=\\\"true\\\">\\n <mi>k</mi>\\n <mo>¯</mo>\\n </mover>\\n <mo>=</mo>\\n <mfenced open=\\\"(\\\" close=\\\")\\\" separators=\\\"|\\\">\\n <mrow>\\n <msub>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <mrow>\\n <mn>1</mn>\\n </mrow>\\n </msub>\\n <mo>,</mo>\\n <mo>…</mo>\\n <mo>,</mo>\\n <msub>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <mrow>\\n <mi>β</mi>\\n <mo>−</mo>\\n <mn>1</mn>\\n </mrow>\\n </msub>\\n </mrow>\\n </mfenced>\\n </math>\\n </jats:inline-formula> so that for each <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M5\\\">\\n <mn>1</mn>\\n <mo>≤</mo>\\n <mi>i</mi>\\n <mo>≤</mo>\\n <mi>β</mi>\\n <mo>−</mo>\\n <mn>1</mn>\\n </math>\\n </jats:inline-formula>, <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M6\\\">\\n <msub>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <mrow>\\n <mi>i</mi>\\n </mrow>\\n </msub>\\n <mo>∈</mo>\\n <msub>\\n <mrow>\\n <mi mathvariant=\\\"double-struck\\\">Z</mi>\\n </mrow>\\n <mrow>\\n <mo>></mo>\\n <mn>0</mn>\\n </mrow>\\n </msub>\\n </math>\\n </jats:inline-formula>, <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M7\\\">\\n <msub>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <mrow>\\n <mi>i</mi>\\n </mrow>\\n </msub>\\n </math>\\n </jats:inline-formula> is coprime to <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M8\\\">\\n <mi>α</mi>\\n </math>\\n </jats:inline-formula> and <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M9\\\">\\n <mi>β</mi>\\n </math>\\n </jats:inline-formula>, and <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M10\\\">\\n <msub>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <mrow>\\n <mi>i</mi>\\n </mrow>\\n </msub>\\n <mo>≡</mo>\\n <mi>i</mi>\\n <mtext> </mtext>\\n <mfenced open=\\\"(\\\" close=\\\")\\\" separators=\\\"|\\\">\\n <mrow>\\n <mi mathvariant=\\\"normal\\\">mod</mi>\\n <mtext> </mtext>\\n <mi>β</mi>\\n </mrow>\\n </mfenced>\\n </math>\\n </jats:inline-formula>. We define the function <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M11\\\">\\n <msub>\\n <mrow>\\n <mi>C</mi>\\n </mrow>\\n <mrow>\\n <mfenced open=\\\"(\\\" close=\\\")\\\" separators=\\\"|\\\">\\n <mrow>\\n <mi>α</mi>\\n <mo>,</mo>\\n <mi>β</mi>\\n <mo>,</mo>\\n <mover accent=\\\"true\\\">\\n <mi>k</mi>\\n <mo>¯</mo>\\n </mover>\\n </mrow>\\n </mfenced>\\n </mrow>\\n </msub>\\n <mo>:</mo>\\n <msub>\\n <mrow>\\n <mi mathvariant=\\\"double-struck\\\">Z</mi>\\n </mrow>\\n <mrow>\\n <mo>></mo>\\n <mn>0</mn>\\n </mrow>\\n </msub>\\n <mo>⟶</mo>\\n <msub>\\n <mrow>\\n <mi mathvariant=\\\"double-struck\\\">Z</mi>\\n </mrow>\\n <mrow>\\n <mo>></mo>\\n <mn>0</mn>\\n </mrow>\\n </msub>\\n </math>\\n </jats:inline-formula> and the sequence <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M12\\\">\\n <mfenced open=\\\"{\\\" close=\\\"}\\\" separators=\\\"|\\\">\\n <mrow>\\n <mi>n</mi>\\n <mo>,</mo>\\n <msub>\\n <mrow>\\n <mi>C</mi>\\n </mrow>\\n <mrow>\\n <mfenced open=\\\"(\\\" close=\\\")\\\" separators=\\\"|\\\">\\n <mrow>\\n <mi>α</mi>\\n <mo>,</mo>\\n <mi>β</mi>\\n <mo>,</mo>\\n <mover accent=\\\"true\\\">\\n <mi>k</mi>\\n <mo>¯</mo>\\n </mover>\\n </mrow>\\n </mfenced>\\n </mrow>\\n </msub>\\n <mrow>\\n <mfenced open=\\\"(\\\" close=\\\")\\\" separators=\\\"|\\\">\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n </mfenced>\\n </mrow>\\n <mo>,</mo>\\n <msubsup>\\n <mrow>\\n <mi>C</mi>\\n </mrow>\\n <mrow>\\n <mfenced open=\\\"(\\\" close=\\\")\\\" separators=\\\"|\\\">\\n <mrow>\\n <mi>α</mi>\\n <mo>,</mo>\\n <mi>β</mi>\\n <mo>,</mo>\\n <mover accent=\\\"true\\\">\\n <mi>k</mi>\\n <mo>¯</mo>\\n </mover>\\n </mrow>\\n </mfenced>\\n </mrow>\\n \",\"PeriodicalId\":43667,\"journal\":{\"name\":\"Muenster Journal of Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2023-09-07\",\"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/10.1155/2023/5159528\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Muenster Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1155/2023/5159528","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

摘要

我们定义函数C α, β,K¯:Z > 0序列n,C α, β,K¯n, c α, β,k ¯
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A New Proof of Rational Cycles for Collatz-Like Functions Using a Coprime Condition
In this paper, we study the bounded trajectories of Collatz-like functions. Fix α , β Z > 0 so that α and β are coprime. Let k ¯ = k 1 , , k β 1 so that for each 1 i β 1 , k i Z > 0 , k i is coprime to α and β , and k i i mod β . We define the function C α , β , k ¯ : Z > 0 Z > 0 and the sequence n , C α , β , k ¯ n , C α , β , k ¯
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