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On the approximation gain for abc-triples

Benne de Weger

TL;DR

This paper extends the abc-triple framework by introducing real, $p$-adic, and combined approximation gains to quantify how well an abc-triple can be approximated through structured factorizations. It defines real enhancements $\mathrm{rrad}_d$, $\mathrm{rag}_d$, and $\mathrm{rpg}_d$, and their extremal versions $\mathrm{rag}$ and $\mathrm{rpg}$, linking them to the $abc$-quality $qu$. It then develops $p$-adic analogues via $\mathrm{mrad}$, $\mathrm{mag}$, $\mathrm{mpg}$ and their single- and multi-prime variants, with a theorem that the multi-$p$-adic gain dominates the real gain, so the combined gain reduces to the multi-$p$-adic gain. The authors illustrate the framework with concrete examples (notably Reyssat’s triple) and provide extensive computational results on massive abc-triple datasets, demonstrating both high gains and the distribution of gains. The work provides a comprehensive toolkit for assessing and comparing abc-triples beyond the classical surrogate surd-based constructions, with online data resources for broader use and SEO-friendly discoverability.

Abstract

The concept of approximation gain was introduced recently by Müller and Taktikos for some abc-triples related to convergents of surds, where there is a relatively large gap between min{a,b,c} and max{a,b,c}. This note proposes a generalization of the concept to all abc-triples, with several variants. Extensive numerical computations are provided.

On the approximation gain for abc-triples

TL;DR

This paper extends the abc-triple framework by introducing real, -adic, and combined approximation gains to quantify how well an abc-triple can be approximated through structured factorizations. It defines real enhancements , , and , and their extremal versions and , linking them to the -quality . It then develops -adic analogues via , , and their single- and multi-prime variants, with a theorem that the multi--adic gain dominates the real gain, so the combined gain reduces to the multi--adic gain. The authors illustrate the framework with concrete examples (notably Reyssat’s triple) and provide extensive computational results on massive abc-triple datasets, demonstrating both high gains and the distribution of gains. The work provides a comprehensive toolkit for assessing and comparing abc-triples beyond the classical surrogate surd-based constructions, with online data resources for broader use and SEO-friendly discoverability.

Abstract

The concept of approximation gain was introduced recently by Müller and Taktikos for some abc-triples related to convergents of surds, where there is a relatively large gap between min{a,b,c} and max{a,b,c}. This note proposes a generalization of the concept to all abc-triples, with several variants. Extensive numerical computations are provided.
Paper Structure (17 sections, 1 theorem, 17 equations, 2 figures, 7 tables)

This paper contains 17 sections, 1 theorem, 17 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The real approximation gain
  3. Introduction
  4. Definitions
  5. Examples
  6. Computational results
  7. The $p$-adic approximation gain
  8. Introduction
  9. Examples
  10. Example 1 -- big gap
  11. Example 2 -- multiple primes
  12. Example 3 -- Reyssat
  13. Definitions
  14. Example 2 revisited
  15. Computational results
  16. ...and 2 more sections

Key Result

Theorem 1

For all $abc$-triples

Figures (2)

  • Figure 1: Distributions of the approximation and power gains.
  • Figure 2: Scatterplots of observed quality, approximation and power gain data.

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Theorem 1
  • Definition 9