Methodological Generalization of the Collatz Sequences to (1 + 2^k )n + S_k(n) with Computational Verification for k = 1 up to k = 42
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| Natura: | Recurso digital |
| Lingua: | inglese |
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2025
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| _version_ | 1866902339848241152 |
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| author | Ammar, HAMDOUS |
| author_facet | Ammar, HAMDOUS |
| contents | <p>Several attempts have been made to generalize the Collatz sequence 3n+1, but<br>many of them produced sequences that lack the essential structural properties<br>of the original Collatz dynamics. Among these, the most promising known<br>generalization is the one proposed in 2022 by Naouel Boulkaboul [2], which<br>takes the form 3n + 3k and leads sequences to converge toward 3<br>k. In this work, we propose a new methodological generalization of the Collatz conjecture<br>based on the transformation (1 + 2k)n + Sk(n), where Sk(n) is a function<br>specifically designed to preserve the singularity and the core behavior of the<br>original sequence. We demonstrate that this generalization retains key features<br>such as singularity patterns and trivial cycles. Furthermore, we conduct a<br>computational verification for k = 1 to k = 42, supporting the conjecture that<br>all such generalized sequences converge to 1. This generalization opens the<br>door to new strategies for understanding and potentially proving the Collatz<br>conjecture in its broader form.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_15530664 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Methodological Generalization of the Collatz Sequences to (1 + 2^k )n + S_k(n) with Computational Verification for k = 1 up to k = 42 Ammar, HAMDOUS Collatz conjecture Generalized Collatz sequences Generalized Collatz conjecture 3n+1 problem Hailstone sequence Discrete dynamics Mathematical singularity Recursive sequences Number theory Structural patterns Binary representation Dynamical systems Conjecture resolution Collatz tree Mathematical logic <p>Several attempts have been made to generalize the Collatz sequence 3n+1, but<br>many of them produced sequences that lack the essential structural properties<br>of the original Collatz dynamics. Among these, the most promising known<br>generalization is the one proposed in 2022 by Naouel Boulkaboul [2], which<br>takes the form 3n + 3k and leads sequences to converge toward 3<br>k. In this work, we propose a new methodological generalization of the Collatz conjecture<br>based on the transformation (1 + 2k)n + Sk(n), where Sk(n) is a function<br>specifically designed to preserve the singularity and the core behavior of the<br>original sequence. We demonstrate that this generalization retains key features<br>such as singularity patterns and trivial cycles. Furthermore, we conduct a<br>computational verification for k = 1 to k = 42, supporting the conjecture that<br>all such generalized sequences converge to 1. This generalization opens the<br>door to new strategies for understanding and potentially proving the Collatz<br>conjecture in its broader form.</p> |
| title | Methodological Generalization of the Collatz Sequences to (1 + 2^k )n + S_k(n) with Computational Verification for k = 1 up to k = 42 |
| topic | Collatz conjecture Generalized Collatz sequences Generalized Collatz conjecture 3n+1 problem Hailstone sequence Discrete dynamics Mathematical singularity Recursive sequences Number theory Structural patterns Binary representation Dynamical systems Conjecture resolution Collatz tree Mathematical logic |
| url | https://doi.org/10.5281/zenodo.15530664 |