Ternary ABC Quality in Number Theory

ISEF Category: Mathematics

Subcategory: Number Theory  ·  Difficulty: Advanced  ·  Setup: Home Setup  ·  Time: 1 to 2 Months

The Hook

Some number patterns hide in plain sight, and then a computer finds something no one expected. This project asks you to turn small integers into triples, measure their abc-quality, and look for unusual cases. You will mix code, pattern spotting, and proof-style thinking. That is a strong combo for a math fair.

What Is It?

The abc-quality of a triple is a way to measure how unusual a relationship is among three numbers that fit an equation like a + b = c. Think of it like a score that rewards triples where the numbers share few common factors and the sum behaves in a surprisingly efficient way. High quality means the triple stands out from the crowd.

This project looks at triples built from ternary expansions, which are just base-three representations of small integers. Base three works like base ten, except each digit can only be 0, 1, or 2. By turning these digit patterns into number triples and then checking their abc-quality, you can search for trends, outliers, and possible new conjectures about how often high-quality triples appear.

Why This Is a Good Topic

This is a good science fair topic because you can generate lots of data with a computer, and the question has a clean mathematical definition. You do not need a lab bench, but you do need careful coding, organized data, and a plan for testing patterns. The topic connects to deep number theory, yet you can still do original work by exploring a new family of triples and comparing their quality distribution to random or baseline cases. You can also learn how mathematicians turn examples into conjectures.

Research Questions

• How does abc-quality change as the size of the ternary integer increases?
• What is the effect of digit pattern type in ternary expansion on the distribution of high-quality triples?
• Does restricting to ternary expansions with few nonzero digits increase the chance of finding high-quality triples?
• To what extent do ternary-derived triples differ from random triples with similar size and gcd structure?
• Which ternary digit patterns produce the largest abc-quality values?
• How does the tail of the abc-quality distribution compare across different ternary subfamilies?
• What is the effect of using different triple-construction rules on the moment estimates of abc-quality?

Basic Materials

• Laptop or desktop computer with Python installed.
• Spreadsheet software for tracking triples and summary statistics.
• Python libraries such as SymPy, NumPy, and Matplotlib.
• Text editor or notebook for writing code and notes.
• Calculator for quick checks of small examples.
• Backup storage or cloud drive for data files.

Advanced Materials

• Laptop or desktop computer with Python, SageMath, or PARI/GP installed.
• High-memory computer or access to a university workstation for larger search ranges.
• Python libraries such as NumPy, SciPy, pandas, Matplotlib, and SymPy.
• SageMath for symbolic checks of gcds, factorization, and number-theory functions.
• Jupyter Notebook for reproducible exploration.
• Version control software such as Git for tracking code changes.

Software & Tools

• Python: Generates triples, computes abc-quality, and runs summary statistics.
• Jupyter Notebook: Keeps code, notes, and plots in one reproducible place.
• SageMath: Helps with factorization, gcd checks, and exact number theory calculations.
• NumPy: Speeds up array-based calculations for large searches.
• Matplotlib: Plots abc-quality distributions and highlights outliers.

Experiment Steps

1. Define exactly how you will turn a ternary expansion into a triple, and write the rule down before coding.
2. Build a small test set by hand so you can confirm your code matches known examples.
3. Decide which variables you will track, such as size, digit count, shared factors, and abc-quality.
4. Create a search plan that scans one family of ternary triples at a time, then stores the results in a clean table.
5. Choose a comparison baseline, such as random triples with similar size, so you can test whether your family behaves differently.
6. Test one conjecture about averages, tails, or moments, then check whether the pattern survives when you expand the data range.

Common Pitfalls

• Using an unclear triple-construction rule, which makes your results impossible to reproduce or compare.
• Forgetting to normalize by gcd, which can inflate abc-quality scores and hide the real pattern.
• Searching too many triples at once without saving metadata, which makes it hard to trace where a high-quality case came from.
• Comparing your ternary family to random triples with very different sizes, which creates a misleading baseline.
• Treating one impressive example as a trend, which leads to weak conjectures and overconfident claims.

What Makes This Competitive

A strong version of this project does more than list high-quality examples. You would define a precise family, compare it to a smart baseline, and test whether the tail behavior looks different from chance. Better still, you would support a conjecture with tables, plots, and a clean argument about why the pattern might hold. A careful search plus a sharp statistical summary can make a math project feel much deeper.

Project Variations

• Change the ternary family to numbers with exactly one nonzero digit block and compare the quality distribution.
• Replace ternary triples with base-four or base-five constructions and test whether the same moment pattern appears.
• Analyze only primitive triples, then compare their abc-quality tail behavior with the full unfiltered sample.

Learn More

• SageMath Documentation: Free documentation and examples for exact arithmetic, factorization, and number theory experiments, found through the official SageMath site.
• MIT OpenCourseWare, Number Theory: Free lecture notes and problem sets that help you build proof skills, found by searching MIT OpenCourseWare for number theory.
• arXiv: Search for preprints on the abc conjecture and computational number theory to see current research directions.
• MathSciNet and zbMATH Open: Search these databases for review articles and paper references on abc-type problems and additive number theory.
• OEIS: Search integer sequences related to ternary expansions, digit sums, and factor patterns to look for related counting sequences.