[Introduction to Number Theory Series Part 1] What Is Number Theory? A 20-Part Roadmap

Number theory studies integers, the most familiar objects in mathematics. But familiarity does not make the subject shallow.

It begins with questions that look simple at first. When does one integer divide another? How do we summarize what two integers share? Why do some numbers break into primes in only one essential way? Why do large computations become easier once we focus on remainders instead of raw numbers?

This post is the starting point of the 20-part series. We will first clarify what number theory looks at, then map how the major ideas connect.

What this post covers

• The recurring questions at the heart of number theory
• The role of the foundational ideas that will reappear throughout the series
• The full 20-part roadmap in stages
• A practical reading path for different kinds of learners

Key terms

• divisibility: the basic relation that asks whether one integer is a multiple of another
• greatest common divisor: the largest positive integer that divides two integers
• prime number: a natural number greater than 1 with no positive divisors other than 1 and itself
• congruence: a way to classify integers by having the same remainder
• Diophantine equation: an equation studied under the restriction that solutions must be integers

What does number theory really study?

Number theory is not just a course about listing divisors and multiples. Its real subject is the structure of relationships among integers.

Questions like these all belong to number theory:

• Why does one number divide another?
• How does the gcd summarize shared structure?
• Why do integers break into primes in an essentially unique way?
• Why do large computations simplify when we move into congruence classes?
• What changes when an equation is required to have integer solutions rather than real ones?

The subject feels rich because the same small set of objects—the integers—keeps revealing deeper layers of structure.

Five recurring questions in number theory

It is often easier to understand a field through its recurring questions than through isolated formulas. This series will repeatedly come back to these five.

1. Does one number divide another?

This is the first relation we need. Once divisibility is in place, ideas like divisors, multiples, quotients, remainders, and the division algorithm all have a natural home.

2. What is the shared structure of two integers?

When we compare two integers, the most important summary value is the gcd. It is not just a computational output. It tells us how much divisor structure the two numbers share.

3. How far can an integer be broken down?

In number theory, primes play the role of basic building blocks. Once you see integers through prime factorization, the structure of divisors, gcds, and lcms becomes much clearer.

4. What becomes easier if we look at remainders?

Sometimes direct computation is hard, but remainder structure is simple. This is where congruence enters. It leads naturally to modular arithmetic, modular inverses, power patterns, and eventually cryptography.

5. Do integer solutions exist?

An equation may have many real solutions but no integer solutions at all. In other cases, the integer restriction reveals a cleaner structure. That is the setting of Diophantine equations.

Why is number theory important?

Number theory is not only a school topic. It also trains mathematical habits that matter far beyond this chapter.

1. It builds mathematical discipline

Number theory repeatedly forces us to read conditions carefully, test patterns on small examples, and then decide what is always true. Questions like “Are these numbers coprime?” or “Is the modulus prime?” are not decoration. They are the difference between a correct statement and a false one.

2. It connects to algorithms

The Euclidean algorithm for the gcd is a classic example. It is not just repeated division. It is a way to reduce a problem to a smaller one quickly and systematically.

3. It leads into cryptography and security

Ideas such as primes, congruence, inverses, and Euler’s totient function form the background of systems like RSA. So number theory is not only abstract. It also reaches into algorithms and real computer systems.

The 20-part roadmap

This series moves in the order basic structure → primes and integer solutions → congruences → core theorems and applications. Each stage depends on the language developed in the previous one.

I. Building the basic structure of integers

Part	Topic	Core idea
1	What is number theory?	scope, recurring questions, and the full roadmap
2	Basic divisibility	divisors, multiples, quotient, remainder
3	The division algorithm	why a=bq+r is the starting form
4	GCD and LCM	shared divisor structure and shared multiple structure
5	The Euclidean algorithm	a fast algorithm for the gcd

Goal: learn how to read the basic relations among integers.

II. Understanding primes and integer solutions

Part	Topic	Core idea
6	Bézout's identity	ax+by=gcd(a,b)
7	Linear Diophantine equations	when ax+by=c has integer solutions
8	Primes	primes, composites, primality intuition
9	Prime factorization	factoring integers into primes
10	Using the fundamental theorem	divisor counts, gcd, and lcm through exponents

Goal: understand both the internal structure of integers and the structure of integer-solution problems.

III. Entering the world of remainders

Part	Topic	Core idea
11	Congruence basics	the meaning of a ≡ b (mod n)
12	Modular arithmetic	addition, subtraction, multiplication, powers
13	Modular inverses	when modular division is possible
14	Linear congruences	solving ax ≡ b (mod n)
15	The Chinese remainder theorem	solving several congruences at once

Goal: develop the habit of replacing large-number computation with remainder structure.

IV. Extending into theorems and applications

Part	Topic	Core idea
16	Fermat's little theorem	powers modulo a prime
17	Euler's totient function and theorem	the general coprime version
18	Arithmetic functions	φ(n), τ(n), σ(n) and related viewpoints
19	Quadratic congruences and residues	a doorway into deeper number theory
20	Applications and next steps	cryptography, algorithms, and future study

Goal: see why number theory remains a deep field and how it connects back to computer science.

How should you read this series?

This series fits readers who:

• know divisors, multiples, and primes but do not yet see how they connect
• want to understand congruence and modular arithmetic structurally rather than by memorization
• want a clean foundation for the number theory behind algorithms and cryptography

A practical reading order is:

1. Parts 1-5 for the basic language of integer structure
2. Parts 6-10 for integer solutions and prime structure
3. Parts 11-15 for congruence and modular arithmetic
4. Parts 16-20 for theorems, arithmetic functions, and applications

Even if your main interest is cryptography, it is usually better not to jump straight into Part 11. The earlier structure makes the later material much more stable.

Wrap-up

Number theory is not a subject of repeated small calculations. It is a way to read the structure of divisibility, primes, remainders, and integer solutions.

This first post mapped the whole series. In the next post, we start from the most basic vocabulary of the subject: divisors, multiples, quotients, and remainders.