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

• URL: http://arxiv.org/abs/2506.10039v1
• Date: Tue, 10 Jun 2025 23:54:56 GMT
• Title: Symbolic Generation and Modular Embedding of High-Quality abc-Triples
• Authors: Michael A. Idowu,
• Abstract summary: We present a symbolic identity for generating integer triples $(a, b, c)$ satisfying $a + b = c$.<n>The construction uses powers of $2$ and $3$ in combination with modular inversion in $mathbbZ/3pmathbbZ$, leading to a parametric identity with residue constraints that yield abc-triples exhibiting low radical values.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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