🔢 mathematics

On the approximation gain for abc-triples

This paper proposes a generalization of the "approximation gain" concept—originally applied to specific $abc$-triples related to surd convergents—to all $abc$-triples and provides extensive numerical evidence for its various forms.

Original authors: Benne de Weger

Published 2026-02-10

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Benne de Weger

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a professional treasure hunter, and you are looking for "Golden Triples"—special sets of three numbers ( $a, b, c$ ) that follow a simple rule ( $a + b = c$ ) but possess a magical property: they are incredibly "dense" with prime factors. In mathematics, these are called abc-triples, and finding ones with high "quality" is like finding a chest overflowing with gold.

For a long time, mathematicians thought these gold chests only appeared in one specific way: when one number was tiny and the other two were massive (like a grain of sand next to two mountains).

This paper, written by Benne de Weger, is like a new set of high-tech goggles that allows treasure hunters to find gold in places they previously thought were just piles of dirt.

1. The Old Way: The "Mountain and Sand" Method

Previously, researchers focused on Real Approximation Gain. This is like looking for a treasure where one number is a tiny speck of dust ( $a$ ) and the other two ( $b$ and $c$ ) are huge. Because the tiny number is so small, it’s easy to see how it "fits" into the gap between the two giants. This is mathematically similar to how we approximate square roots using fractions.

But the problem? Not all gold chests look like that. Some chests have three medium-sized numbers that are all quite large, making them hard to spot with the old "mountain and sand" goggles.

2. The New Way: The "X-Ray" Method (p-adic Gains)

De Weger introduces a brilliant new way to look at these numbers using something called p-adic approximation.

Think of it this way: Imagine you are looking at a giant, complex jigsaw puzzle.

The Real Method is like looking at the puzzle from a distance to see if the pieces fit the overall shape.
The p-adic Method is like using an X-ray to look at the internal structure of the pieces.

Even if a number looks "big" and "heavy" on the outside (in the real world), it might be "light" and "hollow" on the inside because it is divisible by a massive power of a specific prime number (like 2, 3, or 5).

By using these "X-ray goggles," the author can see that a number isn't just a big chunk of stone; it’s actually a highly organized structure of prime factors. This allows him to find "gold" in triples where the numbers are all roughly the same size, which the old methods would have completely missed.

3. The "Combined" Goggles

Finally, the paper suggests that the best way to hunt for treasure is to wear Combined Goggles. This means you don't just look for tiny numbers (Real Gain) or hollow numbers (p-adic Gain); you look for any mathematical reason why the numbers are special.

The Big Picture

The paper isn't just about math formulas; it's about refining our vision.

By splitting the "quality" of these numbers into two parts—Approximation Gain (how well the numbers fit together) and Power Gain (how much "prime power" is packed inside them)—the author provides a much more detailed map of the mathematical landscape.

In short: He has moved us from looking for "big gaps" to looking for "deep patterns," opening up a whole new territory for finding the most precious numbers in mathematics.

Technical Summary: On the Approximation Gain for $abc$-Triples

Author: Benne de Weger
Date: February 10, 2026
Subject: Number Theory (specifically the $abc$-conjecture and Diophantine approximation)

1. The Problem

The paper addresses the decomposition of the "quality" of $abc$-triples. For a coprime triple $(a, b, c)$ where $a + b = c$ , the quality is defined as $qu(a, b, c) = \frac{\log c}{\log \text{rad}(abc)}$ .

Recent work by Müller and Taktikos introduced the concept of approximation gain, which quantifies how much of a triple's high quality is derived from the "closeness" of a rational approximation to a surd (an algebraic number like $k^{1/s}$ ). However, their definition is limited to cases where there is a large gap between the terms (e.g., $a \ll b$ ), which is not representative of all high-quality $abc$-triples. The central problem is to generalize this concept to all $abc$-triples, regardless of the relative sizes of $a, b,$ and $c$ , and regardless of whether they relate to real surds.

2. Methodology

De Weger proposes a multi-faceted framework to redefine approximation gain by shifting the perspective from the real metric to $p$ -adic metrics. The methodology involves:

Real Approximation Gain ($rag$): Generalizing the surd-based approach by looking for the best rational approximation to $(b'/c')^{1/d}$ for various degrees $d$ . This captures the "gap" when one term is significantly smaller than the others in the real sense.
$p$ -adic Approximation Gains: Recognizing that a term can be "small" if it is divisible by a high power of a prime $p$ $p$ . The author utilizes:
- Single $p$ -adic gain ($sag$): Focusing on a single prime $p$ that divides one of the terms.
- Multiple $p$ -adic gain ($mag$): Using the Chinese Remainder Theorem to consider multiple primes simultaneously, effectively treating a product of prime powers as a single $p$ -adic "small" entity.
Lattice Basis Reduction: The author demonstrates that finding these optimal approximations is computationally equivalent to finding short vectors in a two-dimensional lattice (related to continued fractions), which can be solved efficiently using algorithms like LLL or the Euclidean algorithm.
Combined Metrics: To ensure a universal measure, the author defines a combined approximation gain ($cag$) as the maximum of the real and single $p$ -adic gains.

3. Key Contributions

Generalization of Approximation Gain: The paper moves the concept from a niche observation about surds to a robust tool applicable to any $abc$-triple.
Formal Definitions: The introduction of "enhanced radicals" (both real and $p$ -adic) provides a rigorous mathematical basis for splitting quality into two components: approximation gain (how well the terms approximate a root) and power gain (the effect of the prime exponents).
Theorem 1: A formal proof that the multiple $p$ -adic approximation gain is always greater than or equal to the real approximation gain ( $mag \geq rag$ ), establishing the $p$ -adic approach as a superior/encompassing metric.
Comprehensive Taxonomy: The paper categorizes gains into real, single $p$ -adic, multiple $p$ -adic, and combined variants.

4. Results and Computational Findings

The author performed extensive numerical computations on over 14 million $abc$-triples (up to $c < 10^{18}$ ). Key observations include:

Reyssat’s Triple: The famous record-breaking triple ( $2 + 3^{10} \cdot 10^9 = 2^{35}$ ) serves as a benchmark, showing high values in both real and $p$ -adic approximation gains.
Distribution Analysis: The paper provides statistical distributions of these gains, showing how they behave across large datasets.
Performance of Metrics: The $p$ -adic approach successfully "recovers" the high quality of triples that appear to have low real approximation gains (e.g., when $a$ is not much smaller than $b$ ).
Top Triples: The paper identifies the top 10 triples for various gain categories, providing a new way to classify and search for high-quality $abc$-triples.

5. Significance

This work provides a more nuanced lens through which to view the $abc$-conjecture. By decomposing quality into approximation and power gains, researchers can distinguish between triples that are high-quality because of "accidental" prime power distributions (power gain) versus those that arise from deep Diophantine approximation properties (approximation gain). This distinction is crucial for understanding the underlying structure of the $abc$-conjecture and for the development of new algorithms to find record-breaking triples.