A Tour of Computational Mathematical Research

Mathematical Secrets Revealed Through Python

Author

James H. Choi, Ph.D.

Published

January 1, 2026

Introduction

During the “teacher decade” of my life, I had the privilege of teaching many bright high school students. I taught math and physics, but what I enjoyed the most was research mentoring. Many went on to win awards, then on to successful career, having a stint at fabled colleges along the way.

I limited the research subjects to computational math and physics because these are subjects high/middle school students, could actually understand and learn, not some novel cure for cancer for an organ they never heard of before.

And I chose computational research for my students because of the following reasons:

Didn’t require handling/buying dangerous/expensive chemicals/equipment. Access to a computer is all they needed.
No animals, or vegetables for that matter, are harmed in the process.
No need for any lab which tends be available only the night before exams.
No need to obtain IRB’s¹ approval. Students can choose research topics they feel like, not what others approve.
Least affected by environmental factors. No 500-year flood washing away crops students were observing, or unusual clouds blocking their view of the stars for 20 days in a row.

And, yes, the computational experiments progress while students eat or sleep.

I was coming up with research topics on my own, then I ran into two books that really taught and inspired me. They were “Keeping it R.E.A.L.” by Carla Martin (Martin 2011) and “The Computer as Crucible” by Jonathan Borwein (Borwein and Devlin 2009).

Both books gave me both the guidance and the encouragement. Especially the “Further Directions for Undergraduate Research” sections in “Keeping it R.E.A.L.” not only gave me ideas, but more importantly, how to come up with my own ideas. “Crucible” assured me that I was not going strange doing this type of research, but this is a well-established worthy field.

However, those books were intended for college students and up. I found myself translating the materials for high school students. After having played a translator, here I finally write the high school version of the book.

This book is for bright high school students, and even brighter middle school students. This book assumes that you know algebra 1 and 2.² Calculus or geometry is not necessary.

This book also assumes familiarity, but not mastery, of number theory. You need to know what prime numbers and divisors are. If the math prevents you from understanding, please see our companion book for kinder and gentler introduction to the math you need to know. You can also learn it on your own, or with the help of AI. You will be able to ask the right questions and navigate all the way up after finishing this book.

You need to know basic Python. You need to know how to write Python code to check if an integer is prime or not. A companion book is coming to teach you the Python you need to know. You can also learn it on your own, or with the help of AI. You will be able to ask the right questions and navigate all the way up after finishing this book.

But most of all, you also need to have persistence, and curiosity. In this age of AI, you can make up for most gap in knowledge, including the requirements above, with those two traits.

The author

What qualifies me to write this book?

I am a researcher. I worked as a researcher at large US corporate labs for two decades.
I mentored many students to do their own research. Many went on to win awards at diverse fairs from state to ISEF³.
I have been an ISEF judge⁴ for five years.
I have a degree in Physics, EECS⁵, and Mechanical Engineering each.
I am currently a director of research at a company working on AI and machine vision.
I am a certified instructor of Mathematica (Wolfram Language).

Why computational math research?

The reasons I chose to teach this field are multiple. Each one would be a good enough reason for you to pursue the field, as you will see below, but what if one field checks all the boxes? That is the computational math/physics.

Abstract vs Concrete

How computation makes all concepts concrete. From enough concrete examples, you can make abstract generalizations. This is the opposite of the way math is usually taught in school. In school, you are given the abstract generalization first, then you are shown a few examples. In this book, you will be shown many examples first, then you will be able to make your own generalizations and conjectures.

You may feel the list above is just too selfish and calculating. I agree. But we are not done yet. There is another reason that is never mentioned in polite company, not because it is impolite, but because it is a competitive advantage, a trade secret, that won’t be shared.

Computational Research looks great on a college application. Think about it. Colleges look for students that will succeed wildly. They have to crystal ball that future with these young kids who didn’t accomplish anything yet. They have to go by the indicators, the telltale signs. What activity would a future bright star be doing as a middle and high school student? What signs would he/she be showing? Which early sign is more convincing that having a hobby/passion of “using the computing power to make discoveries” and publishing/sharing the results for the public good? Think of the poor college admissions officer who has to read 60 essays a day. While other applicants write about their vague dreams, you will be submitting the list of your publications. If the table were turned, which student would you pick?

Let’s take a break here. I think we have been too selfish. Let’s think about the public image. What public-facing reasons can you give to appear altruistic and well-rounded? Here they are:

Computational research is rewarding on its own merits.
It sharpens the instincts, trains you to ask precise questions.
It gives you the satisfaction of being the first discoverer, and be recognized for it.
It deepens your understanding of the mathematics which advances you in school math.

You may wonder why you weren’t aware of this kind of opportunity that checks so many boxes. First, this opportunity became available only recently as explained below. Second, finding someone to teach these topics to a high school student is not easy. You could ask the AI, and it will answer all your questions. But you must know what to ask in which order. You will be able to ask the right questions and navigate all the way up after finishing this book.

Why do conjectures?

Mathematics has two speeds.

The first is slow and certain: a theorem is proved, then taught, then learned. This is the mathematics of textbooks, moving in one direction — from discovery to you.

The second is fast and uncertain: you notice a pattern, form a guess, then test it against hundreds of examples. This is experimental mathematics, and it moves the other way — from you toward discovery.

Experimental mathematics is not a lesser form of the subject. Some of the deepest results in number theory began as computational observations. The Riemann Hypothesis, which describes the location of prime numbers with eerie precision, is supported by trillions of computed examples and remains unproved more than 150 years after Riemann wrote it down. Goldbach’s conjecture — that every even number greater than 2 is the sum of two primes — has been verified for every even number up to 4,000,000,000,000,000,000 and is still open (Oliveira e Silva et al. 2014).

None of that computing power proved anything. But it told mathematicians where to look, what to believe, and what to try to prove. Computation is a flashlight in a dark room.

The experimental method in mathematics follows a cycle that will repeat throughout this book:

Explore. Compute examples. Plot data. Look for patterns.
Conjecture. State a precise guess: “It seems that …”
Test. Try to break your conjecture with more examples.
Communicate. Write up what you found, whether or not it is proved.

A conjecture that survives serious testing earns the right to be called interesting, even without a proof. A conjecture that fails teaches you something too: it rules out wrong ideas, which is also progress.

You already know enough mathematics to begin. The topics in this book require only algebra. Calculus is not a prerequisite here.

Institutional Review Board. It ensures research meets ethical standards and protects participant welfare.↩︎
That means high-school-level mathematics up to, but not including, trigonometry.↩︎
International Science and Engineering Fair↩︎
I never judged my own students. I switched to different subject when my students qualified.↩︎
Electrical Engineering and Computer Science↩︎

# Introduction {#sec-intro .unnumbered} During the "teacher decade" of my life, I had the privilege of teaching many bright high school students. I taught math and physics, but what I enjoyed the most was research mentoring. Many went on to win awards, then on to successful career, having a stint at fabled colleges along the way. I limited the research subjects to computational **math and physics** because these are subjects high/middle school students, could actually understand and learn, not some novel cure for cancer for an organ they never heard of before. And I chose **computational** research for my students because of the following reasons: 1. Didn't require handling/buying dangerous/expensive chemicals/equipment. Access to a computer is all they needed. 1. No animals, or vegetables for that matter, are harmed in the process. 1. No need for any lab which tends be available only the night before exams. 1. No need to obtain IRB's^[Institutional Review Board. It ensures research meets ethical standards and protects participant welfare.] approval. Students can choose research topics *they* feel like, not what others approve. 1. Least affected by environmental factors. No 500-year flood washing away crops students were observing, or unusual clouds blocking their view of the stars for 20 days in a row. And, yes, the computational experiments progress while students eat or sleep. I was coming up with research topics on my own, then I ran into two books that really taught and inspired me. They were "Keeping it R.E.A.L." by Carla Martin [@martin2011] and "The Computer as Crucible" by Jonathan Borwein [@borwein2009crucible]. Both books gave me both the guidance and the encouragement. Especially the "Further Directions for Undergraduate Research" sections in "Keeping it R.E.A.L." not only gave me ideas, but more importantly, how to come up with my own ideas. "Crucible" assured me that I was not going strange doing this type of research, but this is a well-established worthy field. However, those books were intended for college students and up. I found myself translating the materials for high school students. After having played a translator, here I finally write the high school version of the book. This book is for bright high school students, and even brighter middle school students. This book assumes that you know algebra 1 and 2.^[That means high-school-level mathematics up to, but not including, trigonometry.] Calculus or geometry is not necessary. This book also assumes familiarity, but not mastery, of number theory. You need to know what prime numbers and divisors are. If the math prevents you from understanding, please see our companion book for kinder and gentler introduction to the math you need to know. You can also learn it on your own, or with the help of AI. You will be able to ask the right questions and navigate all the way up after finishing this book. You need to know basic Python. You need to know how to write Python code to check if an integer is prime or not. A companion book is coming to teach you the Python you need to know. You can also learn it on your own, or with the help of AI. You will be able to ask the right questions and navigate all the way up after finishing this book. But most of all, you also need to have persistence, and curiosity. In this age of AI, you can make up for most gap in knowledge, including the requirements above, with those two traits. ## The author What qualifies me to write this book? 1. I am a researcher. I worked as a researcher at large US corporate labs for two decades. 1. I mentored many students to do their own research. Many went on to win awards at diverse fairs from state to ISEF^[International Science and Engineering Fair]. 2. I have been an ISEF judge^[I never judged my own students. I switched to different subject when my students qualified.] for five years. 1. I have a degree in Physics, EECS^[Electrical Engineering and Computer Science], and Mechanical Engineering each. 1. I am currently a director of research at a company working on AI and machine vision. 1. I am a certified instructor of Mathematica (Wolfram Language). ## Why computational math research? {#sec-intro-why} The reasons I chose to teach this field are multiple. Each one would be a good enough reason for you to pursue the field, as you will see below, but what if one field checks all the boxes? That is the computational math/physics. ## Abstract vs Concrete How computation makes all concepts concrete. From enough concrete examples, you can make abstract generalizations. This is the opposite of the way math is usually taught in school. In school, you are given the abstract generalization first, then you are shown a few examples. In this book, you will be shown many examples first, then you will be able to make your own generalizations and conjectures. You may feel the list above is just too selfish and calculating. I agree. But we are not done yet. There is another reason that is never mentioned in polite company, not because it is impolite, but because it is a competitive advantage, a trade secret, that won't be shared. **Computational Research looks great on a college application.** Think about it. Colleges look for students that will succeed wildly. They have to crystal ball that future with these young kids who didn't accomplish anything yet. They have to go by the indicators, the telltale signs. What activity would a future bright star be doing as a middle and high school student? What signs would he/she be showing? Which early sign is more convincing that having a hobby/passion of "*using the computing power to make discoveries*" and publishing/sharing the results for the public good? Think of the poor college admissions officer who has to read 60 essays a day. While other applicants write about their vague dreams, you will be submitting the list of your publications. If the table were turned, which student would you pick? Let's take a break here. I think we have been too selfish. Let's think about the public image. What public-facing reasons can you give to appear altruistic and well-rounded? Here they are: 1. Computational research is rewarding on its own merits. 1. It sharpens the instincts, trains you to ask precise questions. 1. It gives you the satisfaction of being the first discoverer, and be recognized for it. 1. It deepens your understanding of the mathematics which advances you in school math.  You may wonder why you weren't aware of this kind of opportunity that checks so many boxes. First, this opportunity became available only recently as explained below. Second, finding someone to teach these topics to a high school student is not easy. You could ask the AI, and it will answer all your questions. But you must know what to ask in which order. You will be able to ask the right questions and navigate all the way up after finishing this book. ## Why do conjectures? {#sec-intro-experimental} Mathematics has two speeds. The first is slow and certain: a theorem is proved, then taught, then learned. This is the mathematics of textbooks, moving in one direction --- from discovery to you. The second is fast and uncertain: you notice a pattern, form a guess, then test it against hundreds of examples. This is *experimental mathematics*, and it moves the other way --- from you toward discovery. Experimental mathematics is not a lesser form of the subject. Some of the deepest results in number theory began as computational observations. The Riemann Hypothesis, which describes the location of prime numbers with eerie precision, is supported by trillions of computed examples and remains unproved more than 150 years after Riemann wrote it down. Goldbach's conjecture --- that every even number greater than 2 is the sum of two primes --- has been verified for every even number up to 4,000,000,000,000,000,000 and is still open [@oliveira2014]. None of that computing power proved anything. But it told mathematicians where to look, what to believe, and what to try to prove. Computation is a flashlight in a dark room. The experimental method in mathematics follows a cycle that will repeat throughout this book: 1. **Explore.** Compute examples. Plot data. Look for patterns. 2. **Conjecture.** State a precise guess: "It seems that ..." 3. **Test.** Try to break your conjecture with more examples. 4. **Communicate.** Write up what you found, whether or not it is proved. A conjecture that survives serious testing earns the right to be called interesting, even without a proof. A conjecture that fails teaches you something too: it rules out wrong ideas, which is also progress. You already know enough mathematics to begin. The topics in this book require only algebra. Calculus is not a prerequisite here.