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

TL;DR

This paper introduces a symbolic algebraic method for generating high-quality abc-triples with low radical values, combining modular arithmetic and affine transformations to produce both known and novel examples, with potential cryptographic applications.

Contribution

It presents a new symbolic identity and modular embedding technique for generating and analyzing abc-triples, expanding the toolkit for studying the abc conjecture.

Findings

Structured triples with low radical values identified

Novel abc-triples discovered through symbolic methods

Framework suggests applications in cryptography

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.