A Two-Operator Calculus for Arithmetic-Progression Paths in the Collatz Graph
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912446204084224 |
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| author | Angermund, Sebastian |
| author_facet | Angermund, Sebastian |
| contents | A recast of the standard residue-class analysis of the 3x+1 (Collatz) map in terms of two elementary operators on arithmetic progressions. The resulting calculus (i) splits any progression into its even and odd subsequences in a single step, (ii) gives a closed formula for every set of seeds that realises a prescribed parity word, (iii) yields a one line affine invariant that forbids trajectories consisting of infinitely many odd moves, and (iv) reduces the non-trivial-cycle problem to a pair of linear congruences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19115 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Two-Operator Calculus for Arithmetic-Progression Paths in the Collatz Graph Angermund, Sebastian General Mathematics A recast of the standard residue-class analysis of the 3x+1 (Collatz) map in terms of two elementary operators on arithmetic progressions. The resulting calculus (i) splits any progression into its even and odd subsequences in a single step, (ii) gives a closed formula for every set of seeds that realises a prescribed parity word, (iii) yields a one line affine invariant that forbids trajectories consisting of infinitely many odd moves, and (iv) reduces the non-trivial-cycle problem to a pair of linear congruences. |
| title | A Two-Operator Calculus for Arithmetic-Progression Paths in the Collatz Graph |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2506.19115 |