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arXiv · 2506.10039

Symbolic Generation and Modular Embedding of High-Quality abc-Triples

Michael A. Idowu
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Abstract

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.

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Michael A. Idowu. 2025-06-10. Symbolic Generation and Modular Embedding of High-Quality abc-Triples. https://arxiv.org/abs/2506.10039

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