The Shadow Collatz Conjecture: The Journey of All Integers Toward −1
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| Formato: | Recurso digital |
| Lenguaje: | inglés |
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2025
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| _version_ | 1866902197588983808 |
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| author | Miyazawa, Ryosuke |
| author_facet | Miyazawa, Ryosuke |
| contents | <div> <div> <div> <p>This work introduces the “Shadow Collatz Conjecture,” a natural extension of the classical 3n+1 problem to all nonzero integers, including the negative domain. The mapping is defined as follows: if n is even, divide by –2; if n is odd, map to 3n – 1. </p> <p>Through numerical experiments, we verified that every integer from –5×10^7 to +5×10^7 reaches –1 in at most 806 steps. The step-count distribution shows a symmetric, arch-like structure around zero, with a long-tailed histogram and intriguing regularities. </p> <p>Figures and trajectory analyses reveal mirrored dynamics compared to the classical Collatz conjecture, suggesting deeper structural links. Possible future directions include exploring the inverse mapping, generalizing to rational or real numbers, and characterizing convergence time distributions.</p> <p>The PDF contains the full paper with methodology, results, and references. Source code for reproducing the computations and plots will be available on GitHub.</p> </div> </div> </div> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_16754305 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Shadow Collatz Conjecture: The Journey of All Integers Toward −1 Miyazawa, Ryosuke Collatz Conjecture Number Theory Integer Sequences Dynamical Systems Shadow Collatz Conjecture Negative Integers <div> <div> <div> <p>This work introduces the “Shadow Collatz Conjecture,” a natural extension of the classical 3n+1 problem to all nonzero integers, including the negative domain. The mapping is defined as follows: if n is even, divide by –2; if n is odd, map to 3n – 1. </p> <p>Through numerical experiments, we verified that every integer from –5×10^7 to +5×10^7 reaches –1 in at most 806 steps. The step-count distribution shows a symmetric, arch-like structure around zero, with a long-tailed histogram and intriguing regularities. </p> <p>Figures and trajectory analyses reveal mirrored dynamics compared to the classical Collatz conjecture, suggesting deeper structural links. Possible future directions include exploring the inverse mapping, generalizing to rational or real numbers, and characterizing convergence time distributions.</p> <p>The PDF contains the full paper with methodology, results, and references. Source code for reproducing the computations and plots will be available on GitHub.</p> </div> </div> </div> |
| title | The Shadow Collatz Conjecture: The Journey of All Integers Toward −1 |
| topic | Collatz Conjecture Number Theory Integer Sequences Dynamical Systems Shadow Collatz Conjecture Negative Integers |
| url | https://doi.org/10.5281/zenodo.16754305 |