A Playful Introduction to Number Theory
Chapter 1: fundamentals
Common Notation
An introduction to common notation found in Number Theory.
Requirements:
None
Difficulty:
1
Perfect Numbers
Numbers that equal the sum of their proper divisors.
Requirements:
Divisibility
Difficulty:
2
Mersenne Primes
A Deep Dive into the Primes of the Form $2^p - 1$.
Requirements:
Exponents
Difficulty:
2
Fundamental Theorem of Arithmetic
Proving every integer has a unique prime DNA.
Requirements:
Prime Factorization
Difficulty:
2
The Division Algorithm
A formal look at division and remainders.
Requirements:
Arithmetic
Difficulty:
1
The Euclidean Algorithm
A fast and elegant method for finding the Greatest Common Divisor.
Requirements:
Long Division
Difficulty:
2
Chapter 2: arithmetic functions
Sum of Divisors
Introducing a Key Tool, the Sum of Divisors Function.
Requirements:
Prime Factorization
Difficulty:
2
The Sum and Number of Divisors (σ and τ) Revisited
A more formal look at the functions.
Requirements:
Sum of Divisors
Difficulty:
3
Dirichlet Convolution
An operation that combines two arithmetic functions to produce a new function.
Requirements:
Arithmetic Functions
Difficulty:
4
Möbius Inversion
A technique to invert sums over divisors.
Requirements:
Möbius Function
Difficulty:
4
Chapter 3: diophantine equations
The Bézout Identity
Writing GCD of two numbers using only multiplication and addition.
Requirements:
Euclidean Algorithm
Difficulty:
2
Linear Diophantine Equations
Solving equations of the form $ax + by = c$.
Requirements:
The Bézout Identity
Difficulty:
2
Pythagorean Triples
Finding all right-angled triangles with integer sides.
Requirements:
Algebra
Difficulty:
2
Sums of Squares
Which numbers can be written as the sum of two squares? Or four?
Requirements:
Modular Arithmetic
Difficulty:
3
Fermat's Last Theorem
The story of a 350-year-old problem.
Requirements:
General Math
Difficulty:
1
Chapter 4: modular arithmetic
Introduction to Congruences
Congruences explore the number left over after division.
Requirements:
Divisibility
Difficulty:
2
Solving Linear Congruences
Solving for x in $ax ≡ b(mod n)$.
Requirements:
Bézout's Identity
Difficulty:
3
The Chinese Remainder Theorem
Solving systems of simultaneous congruences.
Requirements:
Linear Congruences
Difficulty:
3
Properties of Congruences
Defining Mathematical Operations mod m.
Requirements:
Linear Congruences
Difficulty:
3
Divisibility Tests
Finally, a proof for why divisibility rules work!
Requirements:
Congruences
Difficulty:
2
Euler's Totient Function
Counting numbers coprime to n.
Requirements:
Primes
Difficulty:
3
The Order of an Integer
How many times do you multiply before you get back to 1?
Requirements:
Euler's Totient Function
Difficulty:
3
Primitive Roots
Integers that can generate all others (mod p).
Requirements:
Euler's Theorem
Difficulty:
4
Quadratic Residues
Numbers that have square roots modulo n.
Requirements:
Modular Arithmetic
Difficulty:
4
Chapter 5: cryptography
Introduction to Cryptography
From the Caesar Cipher to modern secrets.
Requirements:
Modular Arithmetic
Difficulty:
1
The RSA Algorithm
The powerhouse behind internet security.
Requirements:
Euler's Theorem
Difficulty:
4
Chapter 6: algebraic
The Law of Quadratic Reciprocity
A surprising and beautiful symmetry labelled as the Golden Theorem of Gauss.
Requirements:
Modular Arithmetic
Difficulty:
4
Chapter 7: analytic
The Prime Number Theorem
How are the primes distributed?
Requirements:
Logarithms
Difficulty:
4
The Riemann Hypothesis
The most famous unsolved problem in mathematics.
Requirements:
Zeta Function
Difficulty:
5
Primes in Arithmetic Progressions
Are there infinite primes of the form $4k + 1$?
Requirements:
Modular Arithmetic, Limits
Difficulty:
5