Elementary Number Theory

Purpose of Course  showclose

A course in elementary number theory concerns itself primarily with simple, arithmetical manipulations of counting numbers: 1, 2, 3, and so forth. These numbers hold a great deal more secrets than one might imagine at first. Despite their apparent simplicity, some of the hardest, most difficult problems in mathematics arise from the study of the theory of numbers. Grade-schoolers can comprehend questions whose solutions evade centuries of investigation. Entirely new fields of mathematical study have grown from research into these questions made possible by number theory.

One of the fundamental objects of study involves the prime numbers. Nearly every integer can be built by multiplying prime numbers, which means that we can solve a large number of problems simply by thinking about them in terms of prime numbers.

Even when a number n is not prime, we can restrict our arithmetic to a set of numbers relatively prime to n and recover many properties of a prime number. A signal for this is related to another important tool of number theory, the greatest common divisor (gcd).

The gcd allows us to solve linear Diophantine equations, which look like the linear equations you studied in precalculus, but restrict their solutions to integer values. Related to the study of linear Diophantine equations is a clockwork mathematics called congruence, where you can assert with a straight face that, for example, 1 + 1 = 0 and not be thrown out of the room.

As the course draws to its finale, we combine prime numbers, relatively prime numbers, and congruence to explain how a deceptively simple problem – factoring an integer into two primes – is in fact so difficult that it can guarantee the security of internet communication, including the credit card number you type when you make an online purchase!

Along the way, we take a few detours for the sake of sightseeing. We examine some problems that fascinated the ancient Greek cult of Pythagoreans – perhaps unto death! This segues nicely into classes of numbers that are obtained easily from the integers, but have some unsettling properties. A recurring theme will focus on how these other classes of numbers preserve the so-called ring properties of the integers. We look at different ways to represent numbers, including the technique of continued fractions, which enjoys some surprising properties. Neither last, nor least, we construct numbers that turn traditional notions of arithmetic on its head, requiring us to reconsider even the definition of our beloved primes!

Some aspects of this course will be experimental: we introduce you to a computer algebra system, Sage, and encourage you to infer patterns and solutions by experimentation.

Course Information  showclose

Welcome to Elementary Number Theory. Below you will find some general information on the course and its requirements.

Primary Resources: This course is composed of a range of different free online materials. However, the course makes primary use of the following materials: Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned materials. You will also need to complete the Final Exam.
 
Note that you will only receive an official grade on your Final Exam. However, in order to adequately prepare for this exam, you will need to work through all of the resources in each unit.
 
In order to pass this course, you will need to earn a 70% or higher on the Final Exam. Your score on the exam will be tabulated as soon as you complete it. If you do not pass the exam, you may take it again.
 
Time Commitment: Completing this course should take you a total of 48 hours. Each unit includes a time advisory that lists the amount of time you are expected to spend on each subunit. These advisories should help you plan your time accordingly. It might be useful to take a look at these time advisories and to determine how much time you have over the next few weeks to complete each unit, and then to set goals for yourself.
 
Tips/Suggestions: As you read or view resources, take careful notes on a separate sheet of paper. These notes will be useful to use as a study guide when you prepare for the Final Exam.

Course Requirements  showclose

In order to take this course, you must:

√    have access to a computer;

√    have continuous broadband Internet access;

√    have the ability/permission to install plug-ins (e.g., Adobe Reader or Flash) and software;

√    have the ability to download and save files and documents to a computer;

√    have the ability to open Microsoft Office files and documents (.doc, .ppt, .xls, etc.);

√    have competency in the English language;

√    have read the Saylor Student Handbook; and

√    Have completed the following courses from “The Core Program” of the Mathematics discipline: MA111: Introduction to Mathematical Reasoning and MA211: Linear Algebra.

Unit Outline show close


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