# Lecture 1: Unique Factorization

### Background Material

• Induction and Recursion

• Sections 1.1-1.2 of the text.
• Maple Getting Started Guide
• Worksheet Management - Go to the "Help" menu and select "Table of Contents" and look for Overview in the "Worksheet Management" folder.

### Topics

• Overview of the course.
• Objectives
• Topics
• Introduction to Maple
• Maple worksheet interface
• Maple help facility
• Basic capabilities of Maple
• Variables, Values, and Names
• Arithmetic
• Symbolic vs. Numeric Computation
• Watch Out
• End with a semi-colon
• Names vs. Values and Evaluation
• Equal vs. Assignment
• Dynamic Scope
• Case Sensitivity
• Maple's assumptions might be different than yours
• Maple is not perfect
• Theorem: p|ab => p|a or p|b, p a prime number
• Proof that sqrt(2) is irrational.
• Investigation of Pythagorean triples.
• Unique Factorization (the fundamental theorem of arithmetic) - the existence of the factorization of an integer into primes follows from the correctness proof of a simple algorithm to compute the prime factorization of an integer, and the uniqueness of the factorization follows from the previous theorem - see ufd.mw for worksheet containing the algorithm.
• Example where unique factorization does not hold: Z[sqrt(10)].

### Assignments

This assignment is a practice assignment not intended to be handed in.