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¯=k1,…,kβ−1 so that for each 1≤i≤β−1, ki∈Z>0, ki is coprime to α and β, and ki≡i mod β. We define the function Cα,β,k¯:Z>0⟶Z>0 and the sequence n,Cα,β,k¯n,Cα,β,k¯2n,⋯...
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Wiley
2023-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2023/5159528 |
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| author | Benjamin Bairrington Nabil Mohsen |
| author_facet | Benjamin Bairrington Nabil Mohsen |
| author_sort | Benjamin Bairrington |
| collection | DOAJ |
| description | 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¯=k1,…,kβ−1 so that for each 1≤i≤β−1, ki∈Z>0, ki is coprime to α and β, and ki≡i mod β. We define the function Cα,β,k¯:Z>0⟶Z>0 and the sequence n,Cα,β,k¯n,Cα,β,k¯2n,⋯ a trajectory of n. We say that the trajectory of n is an integral loop if there exists some N in Z>0 so that Cα,β,k¯Nn=n. We define the characteristic mapping χα,β,k¯:Z>0⟶0,1,…,β−1 and the sequence n,χα,β,k¯n,χα,β,k¯2n,⋯ the characteristic trajectory of n. Let B∈Zβ be a β-adic sequence so that B=χα,β,k¯ini≥0. We say that B is eventually periodic if it eventually has a purely β-adic expansion. We show that the trajectory of n eventually enters an integral loop if and only if B is eventually periodic. |
| format | Article |
| id | doaj-art-fb728591625b4ab7a6b911d1c4d24407 |
| institution | Kabale University |
| issn | 2314-4785 |
| language | English |
| publishDate | 2023-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-fb728591625b4ab7a6b911d1c4d244072025-02-03T01:31:52ZengWileyJournal of Mathematics2314-47852023-01-01202310.1155/2023/5159528A New Proof of Rational Cycles for Collatz-Like Functions Using a Coprime ConditionBenjamin Bairrington0Nabil Mohsen1University of Science and Technology of ChinaUniversity of Science and Technology of ChinaIn 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¯=k1,…,kβ−1 so that for each 1≤i≤β−1, ki∈Z>0, ki is coprime to α and β, and ki≡i mod β. We define the function Cα,β,k¯:Z>0⟶Z>0 and the sequence n,Cα,β,k¯n,Cα,β,k¯2n,⋯ a trajectory of n. We say that the trajectory of n is an integral loop if there exists some N in Z>0 so that Cα,β,k¯Nn=n. We define the characteristic mapping χα,β,k¯:Z>0⟶0,1,…,β−1 and the sequence n,χα,β,k¯n,χα,β,k¯2n,⋯ the characteristic trajectory of n. Let B∈Zβ be a β-adic sequence so that B=χα,β,k¯ini≥0. We say that B is eventually periodic if it eventually has a purely β-adic expansion. We show that the trajectory of n eventually enters an integral loop if and only if B is eventually periodic.http://dx.doi.org/10.1155/2023/5159528 |
| spellingShingle | Benjamin Bairrington Nabil Mohsen A New Proof of Rational Cycles for Collatz-Like Functions Using a Coprime Condition Journal of Mathematics |
| title | A New Proof of Rational Cycles for Collatz-Like Functions Using a Coprime Condition |
| title_full | A New Proof of Rational Cycles for Collatz-Like Functions Using a Coprime Condition |
| title_fullStr | A New Proof of Rational Cycles for Collatz-Like Functions Using a Coprime Condition |
| title_full_unstemmed | A New Proof of Rational Cycles for Collatz-Like Functions Using a Coprime Condition |
| title_short | A New Proof of Rational Cycles for Collatz-Like Functions Using a Coprime Condition |
| title_sort | new proof of rational cycles for collatz like functions using a coprime condition |
| url | http://dx.doi.org/10.1155/2023/5159528 |
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