Symbolic Generation and Modular Embedding of High-Quality abc-Triples
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866908404841185280 |
|---|---|
| author | Idowu, Michael A. |
| author_facet | Idowu, Michael A. |
| contents | We present a symbolic identity for generating integer triples $(a, b, c)$ satisfying $a + b = c$, inspired by structural features of the \emph{abc conjecture}. The construction uses powers of $2$ and $3$ in combination with modular inversion in $\mathbb{Z}/3^p\mathbb{Z}$, leading to a parametric identity with residue constraints that yield abc-triples exhibiting low radical values. Through affine transformations, these symbolic triples are embedded into a broader space of high-quality examples, optimised for the ratio $\log c / \log \operatorname{rad}(abc)$. Computational results demonstrate the emergence of structured, radical-minimising candidates, including both known and novel triples. These methods provide a symbolic and algebraic framework for controlled triple generation, and suggest exploratory implications for symbolic entropy filtering in cryptographic pre-processing. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_10039 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symbolic Generation and Modular Embedding of High-Quality abc-Triples Idowu, Michael A. Cryptography and Security Discrete Mathematics Primary 05A17, Secondary 11D45, 11Y60, 94A60 F.2.1 We present a symbolic identity for generating integer triples $(a, b, c)$ satisfying $a + b = c$, inspired by structural features of the \emph{abc conjecture}. The construction uses powers of $2$ and $3$ in combination with modular inversion in $\mathbb{Z}/3^p\mathbb{Z}$, leading to a parametric identity with residue constraints that yield abc-triples exhibiting low radical values. Through affine transformations, these symbolic triples are embedded into a broader space of high-quality examples, optimised for the ratio $\log c / \log \operatorname{rad}(abc)$. Computational results demonstrate the emergence of structured, radical-minimising candidates, including both known and novel triples. These methods provide a symbolic and algebraic framework for controlled triple generation, and suggest exploratory implications for symbolic entropy filtering in cryptographic pre-processing. |
| title | Symbolic Generation and Modular Embedding of High-Quality abc-Triples |
| topic | Cryptography and Security Discrete Mathematics Primary 05A17, Secondary 11D45, 11Y60, 94A60 F.2.1 |
| url | https://arxiv.org/abs/2506.10039 |