Unit 3: Introduction to Number Theory
This unit is primarily concerned with the set of natural numbers. The axiomatic approach to will be postponed until the unit on recursion and mathematical induction. This unit will help you understand the multiplicative and additive structure of . This unit begins with integer representation: place value. This fundamental idea enables you to completely understand the algorithms we learned in elementary school for addition, subtraction, multiplication, and division of multi-digit integers. The beautiful idea in the Fusing Dots paper will enable you to develop a much deeper understanding of the representation of integers and other real numbers. Then, you will learn about the multiplicative building blocks, the prime numbers. The Fundamental Theorem of Arithmetic guarantees that every positive integer greater than 1 is a prime number or can be written as a product of prime numbers in essentially one way. The Division Algorithm enables you to associate with each ordered pair of non-zero integers - a unique pair of integers - the quotient and the remainder. Another important topic is modular arithmetic. This arithmetic comes from an understanding of how remainders combine with one another under the operations of addition and multiplication. Finally, the unit discusses the Euclidean Algorithm, which provides a method for solving certain equations over the integers. Such equations with integer solutions are sometimes called Diophantine Equations.
Completing this unit should take you approximately 32 hours.
3.1: Place Value Notation
3.2: Prime Numbers
3.2.1: An Infinitude of Primes
3.2.2: Conjectures about Primes
18.104.22.168: The Twin Prime Conjecture
22.214.171.124: Goldbach's Conjecture
126.96.36.199: The Riemann Hypothesis
3.3: Fundamental Theorem of Arithmetic (FTA)
3.4: Modular Arithmetic, the Algebra of Remainders
3.4.1: Division by 3, 9, and 11
3.4.2: Building the Field Z?
3.4.3: Square Roots in Modular Arithmetic
188.8.131.52: The Addition of Remainders
184.108.40.206: The Multiplication of Remainders
3.5: Functions in Number Theory
3.5.1: The Floor and the Ceiling Functions
3.5.2: The Greatest Common Divisor (GCD) and Least Common Multiple (LCM) of Two Integers
3.5.3: The Sigma-Function, Summing Divisors
3.6: The Euclidean Algorithm
3.6.1: Another Look at the Division Algorithm
3.6.2: Solving Ax + By = C over the Integers