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. Gradeschoolers 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 socalled 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
Primary Resources: This course is composed of a range of different free online materials. However, the course makes primary use of the following materials:
 Wissam Raji's An Introductory Course in Elementary Number Theory
 The Sage Foundation's The Sage Notebook
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
√ have access to a computer;
√ have continuous broadband Internet access;
√ have the ability/permission to install plugins (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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Unit 1: Prime Numbers
One of the greatest mathematicians of all time, C. F. Gauss, remarked that “Mathematics is the queen of the sciences, and number theory is the queen of mathematics”; we would modify this only by adding that, “prime numbers are the queen of number theory.” Indeed, the major goal of this unit is to introduce you to the prime numbers, a powerful tool that we rely on repeatedly throughout the course. We have no doubt that, as you close the metaphorical book on the final unit, you will agree with our addendum to Gauss’s observation.
Unit 1 Learning Outcomes show close
As this is a theoretical course, we are interested more in precision and properties than in computation. It is very easy to be led astray by vague notions – historically, incorrect assertions in number theory are due precisely to this fact – so we need to make our ideas precise. You will find much of the material in the first two sections familiar, as you have seen them in MA111 and MA231. By and large, the numbers that we study in elementary number theory are those obtainable by performing simple operations on the integers, so we start with a review of the structure of the integers: their ring properties, their orderings, and mathematical induction. The property of being prime, called primality, is related to factorization, which is related to divisibility. This requires us to precede our investigation of primality with a precise investigation of what it means to divide two integers; it may surprise you that, in this course, we almost never pay attention to the quotient, but focus instead on the remainder.
With the review out of the way, we turn to the Fundamental Theorem of Arithmetic, which states that every integer larger than 2 can be written as a product of prime numbers in exactly one way. We show this in two different ways, then take a look at the question, “Just how many primes are there, anyway?” It turns out that there are infinitely many, but whether this is more interesting than the reason why is debatable. Mathematicians like to say that if God had a book that contained the most elegant proofs ever written, Euclid’s solution to this question would surely take its place among them. We also ask, “How many prime numbers can we find, up to a certain size?” Alas, a proof would take us beyond elementary number theory, so we omit it – but the result is still worth discussing.
 1.1 Integers and Induction

1.1.1 Ring Properties of the Integers
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Chapter 1: Introduction and Section 1.1: Algebraic Operations with Integers”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Chapter 1: Introduction and Section 1.1: Algebraic Operations with Integers” (PDF)
Instructions: A ring is a set with two operations, addition and multiplication, that satisfy the ordinary rules of addition and multiplication. Read the introduction to chapter 1 and “Section 1.1: Algebraic Operations with Integers” on pages 7–10 to refamiliarize yourself with the properties of these operations. Despite its familiarity, don't rush through it.
Reading these sections and taking notes should take approximately 15 minutes.  Assessment: The Saylor Foundation’s “Justification that the Additive Inverse of any Element Is Unique”
Link: The Saylor Foundation’s “Justification that the Additive Inverse of any Element Is Unique” (PDF)
Instructions: This first assessment gives a gentle introduction. We have outlined a proof that the additive inverse of any element of a ring is unique. This proof applies not only to the integers, but to any ring. We have left some spaces blank. Fill in those spaces with the appropriate property listed in the reading. You need not be quite so pedantic when completing most of the assessments in this course, but the problem itself illustrates how often we glide over the seemingly obvious and why you should try to be careful when doing assessments like these.
Completing this assessment should take approximately 15 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Rings and Fields Exercise”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Rings and Fields Exercise” (PDF)
Instructions: Try to do Exercise 1 on page 10.
After attempting Exercise 1 assigned above, discuss your solution in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assignment should take approximately 15 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Chapter 1: Introduction and Section 1.1: Algebraic Operations with Integers”

1.1.2 Well Orderings
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.2.1: The Well Ordering Principle”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.2.1: The Well Ordering Principle” (PDF)
Instructions: Read “Section 1.2.1: The Well Ordering Principle” on page 11.
Reading this section and taking notes should take less than 15 minutes.  Reading: Citizendium: “Countable Set: Rational Numbers”
Link: Citizendium: “Countable Set: Rational Numbers” (HTML)
Instructions: Read the Section on Rational Numbers carefully, as you will need it for the next assessment.
Reading this webpage, taking notes, and studying the proof should take approximately 15 minutes.
This resource is licensed under a Creative Commons AttributionShareAlike 3.0 Unported License. It is attributed to Citizendium.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.2.1: The Well Ordering Principle”

1.1.3 Mathematical Induction
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.2.2: The Pigeonhole Principle and Section 1.2.3: The Principle of Mathematical Induction”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.2.2: The Pigeonhole Principle and Section 1.2.3: The Principle of Mathematical Induction” (PDF)
Instructions: Read “Section 1.2.2: The Pigeonhole Principle” and “Section 1.2.3: The Principle of Mathematical Induction” on pages 11–14.
Reading these sections, taking notes, and studying the examples should take approximately 15 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Mathematical Induction Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Mathematical Induction Exercises” (PDF)
Instructions: Try to do Exercises 3 and 6 on page 14.
After attempting Exercises 3 and 6 assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take between 15 minutes and 1 hour, depending on your comfort level with the material.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.2.2: The Pigeonhole Principle and Section 1.2.3: The Principle of Mathematical Induction”

1.1.4 Geometric Numbers
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 3.1: Geometric Numbers”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 3.1: Geometric Numbers” (PDF)
Instructions: Read “Section 3.1: Geometric Numbers” on pages 57–59.
Reading this section, taking notes, and studying the examples should take approximately 15 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Geometric Number Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Geometric Number Exercises” (PDF)
Instructions: Try to do Exercises 1 and 2 on page 59.
After attempting Exercises 1 and 2 assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take less than 30 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 3.1: Geometric Numbers”
 1.2 The Division Algorithm

1.2.1 Integer Divisibility
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.3.1: Integer Divisibility”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.3.1: Integer Divisibility” (PDF)
Instructions: Read “Section 1.3.1: Integer Divisibility” on pages 1516.
Reading this section, taking notes, and studying the examples should take less than 15 minutes.  Assessment: The Saylor Foundation’s “Theorem 4: Sketch of Proof of Equation (1.6)”
Link: The Saylor Foundation’s “Theorem 4: Sketch of Proof of Equation (1.6)” (PDF)
Instructions: Use induction to show that equation (1.6) on page 16 of An Introductory Course in Elementary Number Theory is true. You can use the sketch of the proof, provided at the link.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Divisibility Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Divisibility Exercises” (PDF)
Instructions: Try to do Exercises 1, 4, 6, 7, 11, and 12 on pages 1718. For 6 and 7, use the forms of even and odd numbers given in Example 3 on page 15.
After attempting the divisibility exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 30 minutes, depending on your comfort level with the material.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.3.1: Integer Divisibility”

1.2.2 The Division Algorithm
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.3.2: The Division Algorithm”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.3.2: The Division Algorithm” (PDF)
Instructions: Read “Section 1.3.2: The Division Algorithm” on pages 16–17.
Reading this section and taking notes should take less than 15 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Division Algorithm Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Division Algorithm Exercises” (PDF)
Instructions: Try to do Exercises 2 and 3 on page 17.
After attempting the division algorithm exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 15 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Divisibility Exercises 5 and 8”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Divisibility Exercises 5 and 8” (PDF)
Instructions: Try to do Exercises 5 and 8 on page 17. It will help to divide this problem into three cases, organized by the remainder of division of m by 3. For each case, write m in a form similar to that used in Example 3 for even and odd numbers.
After attempting the divisibility exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 15 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.3.2: The Division Algorithm”
 1.3 Definition and Characterization of a Prime Number

1.3.1 Definition via Divisibility (Traditional Definition) and the Sieve of Eratosthenes
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 2.1: The Sieve of Eratosthenes”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 2.1: The Sieve of Eratosthenes” (PDF)
Instructions: Read “Section 2.1: The Sieve of Eratosthenes” on pages 35–37.
Reading this section, taking notes, and studying the examples should take approximately 15 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  The Sieve of Eratosthenes Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  The Sieve of Eratosthenes Exercises” (PDF)
Instructions: Try to do Exercises 1, 3, and 4 on page 37. Exercise 3 can be resolved by a factorization formula. To best use the hint for Exercise 4, try proving the contrapositive.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 15 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 2.1: The Sieve of Eratosthenes”

1.3.2 Definition via Division (Euclid’s Criterion)
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 2.2: Alternate Definition of Prime Number”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 2.2: Alternate Definition of Prime Number” (PDF)
Instructions: Read “Section 2.2: Alternate Definition of Prime Number” on pages 38–39.
Reading this section, taking notes, and studying the examples should take approximately 15 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 2.2: Alternate Definition of Prime Number”
 1.4 Prime Numbers Are the Building Blocks of Integers!

1.4.1 The Fundamental Theorem of Arithmetic: Traditional Proof
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 2.4: Introduction and Section 2.4.1: The Fundamental Theorem of Arithmetic”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 2.4: Introduction and Section 2.4.1: The Fundamental Theorem of Arithmetic” (PDF)
Instructions: Read the introduction to “Section 2.4” on page 41, and “Section 2.4.1: The Fundamental Theorem of Arithmetic” on pages 41–44.
Reading these sections, taking notes, and studying the examples should take approximately 15 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Prime Factorization Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Prime Factorization Exercises” (PDF)
Instructions: Try to do Exercises 1, 2, and 3 on page 46.
After attempting the prime factorization exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 30 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 2.4: Introduction and Section 2.4.1: The Fundamental Theorem of Arithmetic”

1.4.2 Sage Lab: Introduction to Sage
 Web Media: The Sage Foundation’s “The Sage Notebook”
Link: The Sage Foundation’s “The Sage Notebook” (HTML)
Instructions: Visit The Sage Notebook website, create an account for yourself. To create an account you will need to click on the Google, Yahoo!, or OpenID buttons.
After you’ve created an account, go to the Sage guided tour found here and work through the following sections:
 Assignment, Equality, and Arithmetic
 Getting Help
 Functions, Indentation, and Counting
 Basic Algebra and Calculus (Stop once you get to “Solving Differential equations.”)
 Plotting
 Some Common Issues with Functions
 Basic Rings
 Linear Algebra (Stop once you get to “Matrix spaces.”)
This resource is licensed under a Creative Commons Attribution 3.0 Unported License. Attributions can be found here.
 Web Media: The Sage Foundation’s “The Sage Notebook”
 1.5 How Many Prime Numbers Are There? Absolute and Relative Measures

1.5.1 There Are Infinitely Many Primes
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 2.3: The infinitude of Primes”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 2.3: The infinitude of Primes” (PDF)
Instructions: Read “Section 2.3: The Infinitude of Primes” on pages 3940.
Reading this section, taking notes, and studying the examples should take approximately 5 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  The Infinitude of Primes Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  The Infinitude of Primes Exercises” (PDF)
Instructions: Try to do Exercises 1, 3, and 4 on pages 4041.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 30 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 2.3: The infinitude of Primes”

1.5.2 Sage Lab: “Relatively Speaking, How Many Primes Are There?”
 Web Media: The Saylor Foundation’s “Sage Lab: Relatively Speaking, How Many Primes Are There?”
Link: The Saylor Foundation’s “Sage Lab: Relatively Speaking, How Many Primes Are There?” (Sage)
Instructions: Download the linked set of labs. Upload the first one (1.5.2SageWS1.sws) to the Sage website where you created an account (subunit 1.4.2). Work through the lab, trying to guess a formula for π(x).
Completing this assignment should take approximately 15 minutes.
 Web Media: The Saylor Foundation’s “Sage Lab: Relatively Speaking, How Many Primes Are There?”

1.5.3 The Prime Number Theorem (Without Proof) and Some Famous Conjectures Regarding Prime Numbers
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 2.8: Theorems and Conjectures Involving Prime Numbers”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 2.8: Theorems and Conjectures Involving Prime Numbers” (PDF)
Instructions: Read “Section 2.8: Theorems and Conjectures Involving Prime Numbers” on pages 5456.
Reading this section, taking notes, and studying the examples should take less than 15 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 2.8: Theorems and Conjectures Involving Prime Numbers”

Unit 2: The Greatest Common Divisor
Even when an integer is not prime, it can still behave in a primelike fashion with many other integers. We call two integers that act this way relatively prime. This concept is as important to elementary number theory as the notion of prime numbers, and it depends entirely on the greatest common divisor of the two integers in question, abbreviated as their gcd. A surprisingly efficient method to compute the gcd is due to the same Euclid who showed us earlier that there are infinitely many primes, and we will study it here in some detail.
Unit 2 Learning Outcomes show close
We next turn to the topic of linear Diophantine equations. These are basically the same linear equations that you studied in precalculus, but with a twist: we only want integer solutions. This greatly restricts what we can do, but the gcd provides a criterion for when a linear Diophantine equation can be solved, and the Euclidean algorithm – which we need to compute the gcd anyway – provides a splendid technique for solving them. This leads us to Bezout’s identity, which “turns the tables” on these relationships and provides a powerful tool for subsequent units.
 2.1 The Greatest Common Divisor

2.1.1 Definition and Elementary Properties
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.5: The Greatest Common Divisor”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.5: The Greatest Common Divisor” (PDF)
Instructions: Read “Section 1.5: The Greatest Common Divisor” on pages 2124.
Reading this section, taking notes, and studying the examples should take approximately 30 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  GCD Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  GCD Exercises” (PDF)
Instructions: Try to do Exercises 1, 3, 4, 5, and 6 on page 25.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 1 hour.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.5: The Greatest Common Divisor”

2.1.2 Dirichlet’s Theorem
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory: Section 2.4.1: The Fundamental Theorem of Arithmetic and Section 2.4.2: More on the Infinitude of Primes”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory: Section 2.4.1: The Fundamental Theorem of Arithmetic and Section 2.4.2: More on the Infinitude of Primes” (PDF)
Instructions: Reread “Section 2.4.1: The Fundamental Theorem of Arithmetic” on pages 41–44. Then read “Section 2.4.2: More on the Infinitude of Primes” on pages 4546.
Reading these sections, taking notes, and studying the examples should take approximately 1 hour.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Infinitude of Primes Exercise”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Infinitude of Primes Exercise” (PDF)
Instructions: Try to do Exercise 4 on page 46.
After attempting the exercise assigned above, discuss your solution in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 15 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory: Section 2.4.1: The Fundamental Theorem of Arithmetic and Section 2.4.2: More on the Infinitude of Primes”
 2.2 The Euclidean Algorithm

2.2.1 Statement of the Algorithm and Some Examples
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.6: The Euclidean Algorithm”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.6: The Euclidean Algorithm” (PDF)
Instructions: Read “Section 1.6: The Euclidean Algorithm” on pages 2528. Be sure to follow the examples carefully!
Reading this section, taking notes, and studying the examples should take approximately 30 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Euclidean Algorithm Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Euclidean Algorithm Exercises” (PDF)
Instructions: Try to do Exercises 2, 3, 4, and 5 on pages 28–29.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 1 hour.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.6: The Euclidean Algorithm”

2.2.2 “How Much Time Do I Have To Waste On This?” An Analysis of the Algorithm’s Efficiency
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.7.1: Lame’s Theorem”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.7.1: Lame’s Theorem” (PDF)
Instructions: Read “Section 1.7.1: Lame’s Theorem” on pages 2931.
Reading this section, taking notes, and studying the examples should take approximately 30 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Upper Bound Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Upper Bound Exercises” (PDF)
Instructions: Try to do Exercises 3 and 4 on page 34.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 30 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.7.1: Lame’s Theorem”

2.2.3 Binet’s Formula and the Golden Ratio
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.7.2: Binet’s Formula”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.7.2: Binet’s Formula” (PDF)
Instructions: Read “Section 1.7.2: Binet’s Formula” on pages 31–34.
Reading this section, taking notes, and studying the examples should take approximately 30 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Binet’s Formula Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Binet’s Formula Exercises” (PDF)
Instructions: Try to do Exercises 5 and 6 on page 34.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 30 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 1.7.2: Binet’s Formula”

2.2.4 The Lucas Sequence (Optional)
 Optional Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Lucas Sequence Exercise”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Lucas Sequence Exercise” (PDF)
Instructions: Try to do Exercise 7 on page 34.
After attempting the exercise assigned above, discuss your solution in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 1 hour.
 Optional Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Lucas Sequence Exercise”
 2.3 Linear Diophantine Equations

2.3.1 Definition and Criterion for Solvability
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 2.6: Linear Diophantine Equations”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 2.6: Linear Diophantine Equations” (PDF)
Instructions: Read “Section 2.6: Linear Diophantine Equations” on pages 4951.
Reading this section, taking notes, and studying the examples should take approximately 15 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Linear Diophantine Equation Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Linear Diophantine Equation Exercises” (PDF)
Instructions: Try to do Exercises 1, 2, and 4 on page 51.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 1 hour.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 2.6: Linear Diophantine Equations”

2.3.2 Sage Lab: “Visualizing the Solutions to Linear Diophantine Equations”
 Web Media: The Saylor Foundation’s Sage Lab: “Visualizing Solutions to Linear Diophantine Equations”
Link: The Saylor Foundation’s Sage Lab: “Visualizing Solutions to Linear Diophantine Equations” (Sage)
Instructions: Download the linked set of labs. Upload the second one (2.3.2SageWS2.sws) to the Sage website where you created an account (subunit 1.4.2). Work through the lab carefully.
Completing this assignment should take approximately 30 minutes.
 Web Media: The Saylor Foundation’s Sage Lab: “Visualizing Solutions to Linear Diophantine Equations”

Unit 3: Congruence, with Applications
If you weren't already convinced of our claim in Unit 1 that the remainder is more interesting and useful for number theory than quotients, then this unit will overwhelm you with evidence. Congruence begins as an innocentlooking relationship based on divisibility, which itself is a property of the remainder. We quickly show that this superficial simplicity masks a tool of incredible power. It determines an equivalence relation, and captures the ring properties of the integers – but not some other properties of the integers! We call the concomitant equivalence classes residues, and show that it suffices to perform any arithmetic on the residues by... computing the remainders!
Unit 3 Learning Outcomes show close
With some theory out of the way, we turn immediately to applications. Many interesting problems in the real world can be stated in terms of linear congruence relations, a disguised form of our recentlyacquired friend, linear Diophantine equations. This provides a method for solving not only linear congruence relation, but also systems of linear congruence relations. This latter problem is an ancient one, used in practical situations by both Chinese and Indian mathematicians, and enjoys a property called the Chinese Remainder Theorem. We prove this theorem two different ways; unlike the Fundamental Theorem of Arithmetic, each proof gives us a practical method for solving the problems.
Finally, we return to the question of primality, studying congruence in the context of a prime number. In this case, the residues enjoy the properties not only of a ring, but of a field, which leads to a tool for primality testing.
 3.1 Congruence

3.1.1 Definition, Examples, and Ring Properties of Congruence
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 4.1: Introduction to Congruences”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 4.1: Introduction to Congruences” (PDF)
Instructions: Read “Section 4.1: Introduction to Congruences” on pages 77–81, through “Example 29.” Stop when you get to “Theorem 25.”
Reading this section, taking notes, and studying the examples should take approximately 30 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Congruence Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Congruence Exercises” (PDF)
Instructions: Try to do Exercises 14 and 611 on page 83.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 1 hour.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 4.1: Introduction to Congruences”

3.1.2 Review of Equivalence Relations
 Reading: Elias Zakon’s “Mathematical Analysis: Volume I”
Link: Elias Zakon’s “Mathematical Analysis: Volume I” (PDF)
Instructions: Read pages 1214, starting from the heading “Definition 4.”
Reading this section, taking notes, and studying the examples should take approximately 15 minutes.  Assessment: Elias Zakon’s “Mathematical Analysis: Volume I  Equivalence Relation Exercise”
Link: Elias Zakon’s “Mathematical Analysis: Volume I  Equivalence Relation Exercise” (PDF)
Instructions: Try to do Exercise 8 on page 15.
After attempting the exercise assigned above, discuss your solution in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take from 15 minutes to 1 hour, depending on your comfort level with the material.
 Reading: Elias Zakon’s “Mathematical Analysis: Volume I”

3.1.3 Congruence as an Equivalence Relation
 Assessment: The Saylor Foundation's “Justification that Congruence Is a Symmetric Relation”
Link: The Saylor Foundation's “Justification that Congruence Is a Symmetric Relation” (PDF)
Instructions: (1) Show that congruence is an equivalence relation, by showing that it satisfies the reflexive, symmetric, and transitive properties. (2) Explain what Theorem 2 of Elias Zakon’s “Mathematical Analysis: Volume I” implies about congruence classes.
Completing this assessment should take approximately 30 minutes.
 Assessment: The Saylor Foundation's “Justification that Congruence Is a Symmetric Relation”

3.1.4 Familiar Properties Not Satisfied by Congruence
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory” (PDF)
Instructions: Read pages 8182, starting immediately after “Example 29.”
Reading this section, taking notes, and studying the examples should take approximately 15 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory”

3.1.5 Residue Systems
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 4.2.1: Residue Systems”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 4.2.1: Residue Systems” (PDF)
Instructions: Read “Section 4.2.1 Residue Systems on pages 8485.
Reading this section, taking notes, and studying the examples should take approximately 15 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Residue Systems Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Residue Systems Exercises” (PDF)
Instructions: Try to do Exercises 1 and 2 on page 86.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 15 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 4.2.1: Residue Systems”

3.1.6 Sage Lab: “Residue Arithmetic”
 Web Media: The Saylor Foundation’s Sage Lab: “Residue Arithmetic”
Link: The Saylor Foundation’s Sage Lab: “Residue Arithmetic” (Sage)
Instructions: Download the linked set of labs. Upload the third one (3.1.7SageWS3.sws) to the Sage website where you created an account (subunit 1.4.2). Work through the lab carefully.
Completing this assignment should take approximately 30 minutes.
 Web Media: The Saylor Foundation’s Sage Lab: “Residue Arithmetic”

3.1.7 The Secret Life of Linear Congruence Relations (Linear Diophantine Equations)
A linear congruence relation has the form ax ≡ b (mod m), where aand bare known constants, while xis the unknown variable. By the definitions of congruence and divisibility, we must be able to find an integer qsuch that ax – b = mq. Let y = q, and we can rewrite this equation as
ax + my = b
called a Diophantine equation. (The variables xand yare to the first degree, so we call it a linear Diophantine equation.) Diophantine equations have a long, storied history in number theory, and some of number theories most important problems have been Diophantine equations. We would like to solve these sorts of equations, in part because, for now, we should like to solve linear congruence relations, but later there will be other applications, as well. As you will see, you already knowhow to solve these equations! Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 2.6: Linear Diophantine Equations”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 2.6: Linear Diophantine Equations” (PDF)
Instructions: Reread “Section 2.6 Linear Diophantine Equations on pages 49–51. If you are still familiar with this material, you can skip to the next reading.
Rereading this section, taking notes, and studying the examples should take approximately 15 minutes.  Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Linear Congruences”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Linear Congruences” (PDF)
Instructions: Read “Section 4.3 Linear Congruences” on pages 8182.
Reading this section, taking notes, and studying the examples should take approximately 15 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Linear Congruence Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Linear Congruence Exercises” (PDF)
Instructions: Try to do Exercises 14 on pages 8889.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 1 hour.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 2.6: Linear Diophantine Equations”
 3.2 The Other Sun Tzu’s “Art of War”

3.2.1 The Chinese Remainder Theorem
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Introduction to Section 4.4: The Chinese Remainder Theorem and Section 4.4.1: Direct Solution”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Introduction to Section 4.4: The Chinese Remainder Theorem and Section 4.4.1: Direct Solution” (PDF)
Instructions: Read the introduction to “Section 4.4: The Chinese Remainder Theorem” and all of “Section 4.4.1: Direct Solution” on pages 8991.
Reading these sections, taking notes, and studying the examples should take approximately 30 minutes.  Assessment: Wissam Raji’s “An Introductory Coursein Elementary Number Theory  Chinese Remainder Theorem Direct Solution Exercises”
Link: Wissam Raji’s “An Introductory Coursein Elementary Number Theory  Chinese Remainder Theorem Direct Solution Exercises” (PDF)
Instructions: Try to do Exercises 13 on page 91.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 30 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Introduction to Section 4.4: The Chinese Remainder Theorem and Section 4.4.1: Direct Solution”

3.2.2 Incremental Solution
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 4.4.2: Incremental Solution”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 4.4.2: Incremental Solution” (PDF)
Instructions: Read “Section 4.4.2: Incremental Solution” on pages 9192.
Reading this section, taking notes, and studying the examples should take approximately 15 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Chinese Remainder Theorem Incremental Solution Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Chinese Remainder Theorem Incremental Solution Exercises” (PDF)
Instructions: Try to do Exercises 13 on page 92.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 30 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 4.4.2: Incremental Solution”

3.2.3 Sage Lab: “Chinese Remainders and Polynomial Factorization”
 Web Media: The Saylor Foundation’s Sage Lab: “Chinese Remainders and Polynomial Factorization”
Link: The Saylor Foundation’s Sage Lab: “Chinese Remainders and Polynomial Factorization” (Sage)
Instructions: Download the linked lab. Upload the fourth one (3.2.3SageWS4.sws) it to the Sage website where you created an account (subunit 1.4.2). Work through the lab carefully.
Completing this assignment should take approximately 30 minutes.
 Web Media: The Saylor Foundation’s Sage Lab: “Chinese Remainders and Polynomial Factorization”
 3.3 Primality Testing and Fermat’s Little Theorem

3.3.1 Field Properties of Congruence Modulo p
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Introduction to Section 4.5: Field Properties of Residues Modulo a Prime, and a Primality Test and Section 4.5.1: When Is a System of Residues a Field?”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Introduction to Section 4.5: Field Properties of Residues Modulo a Prime, and a Primality Test and Section 4.5.1: When Is a System of Residues a Field?” (PDF)
Instructions: Read the introduction to “Section 4.5: Field Properties of Residues Modulo a Prime, and a Primality Test” and all of “Section 4.5.1: When Is a System of Residues a Field? on pages 9396.
Reading these sections, taking notes, and studying the examples should take approximately 30 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Introduction to Section 4.5: Field Properties of Residues Modulo a Prime, and a Primality Test and Section 4.5.1: When Is a System of Residues a Field?”

3.3.2 Fermat’s Little Theorem as a Test for Primality
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 4.5.2: Fermat’s Little Theorem as a Primality Test”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 4.5.2: Fermat’s Little Theorem as a Primality Test” (PDF)
Instructions: Read “Section 4.5.2: Fermat’s Little Theorem as a Primality Test” on page 96.
Reading this section, taking notes, and studying the examples should take less than 15 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Fermat’s Little Theorem Exercise”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Fermat’s Little Theorem Exercise” (PDF)
Instructions: Try to do Exercise 1 on page 96.
After attempting the exercise assigned above, discuss your solution in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 15 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 4.5.2: Fermat’s Little Theorem as a Primality Test”

Unit 4: Numbers that Lack Integrity
This unit focuses primarily on objects that solve mathematical problems, but are neither integers, arithmetic combinations of integers, nor even algebraic combinations of integers. The first topic we owe to the Pythagoreans; their wellknown theorem of right triangles led them to a discovery they actually found repugnant: the existence of socalled irrational numbers.
Unit 4 Learning Outcomes show close
As you know, some numbers are not merely irrational; they are imaginary. Our last set of interesting numbers consists of the Gaussian integers, complex numbers whose real and imaginary parts are integers. They resemble the integers in many ways – ring properties, a way to divide with remainder – but they differ in one very important way: the notion of a prime number. We don't go into much depth on that here, but merely introduce the problem, then refine it as the course progresses.
By the 19^{th} century, mathematicians’ attitudes toward numbers that didn't fit a certain scheme were somewhat more accepting, so when they encountered numbers that weren't algebraic, they apparently felt that these numbers in some sense rose above the others, and called them transcendental. Any spiritual or religious connotation is purely coincidental.
 4.1 Rational Numbers, Irrational Numbers, and Their Existence

4.1.1 Field Properties of the Rational Numbers
 Assessment: Show that the set Q of rational numbers forms a field under their ordinary addition and multiplication.
Instructions: Show that the set Q of rational numbers forms a field under their ordinary addition and multiplication.
Completing this assessment should take approximately 15 minutes.
 Assessment: Show that the set Q of rational numbers forms a field under their ordinary addition and multiplication.

4.1.2 The Pythagoreans’ Unpleasant Surprise: Irrational Numbers
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 3.2: Irrational Numbers”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 3.2: Irrational Numbers” (PDF)
Instructions: Read “Section 3.2: Irrational Numbers” on pages 5961.
Reading this section, taking notes, and studying the examples should take approximately 15 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Irrational Numbers Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Irrational Numbers Exercises” (PDF)
Instructions: Try to do Exercises 1 and 2 on page 61.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 30 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 3.2: Irrational Numbers”
 4.2 Gaussian Integers

4.2.1 “Ring Properties” of the Gaussian Integers
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Introduction to Section 3.3: Gaussian Integers and Section 3.3.1: Ring Properties of Z[i]”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Introduction to Section 3.3: Gaussian Integers and Section 3.3.1: Ring Properties of Z[i]” (PDF)
Instructions: Read the introduction to “Section 3.3: Gaussian Integers” and all of “Section 3.3.1: Ring Properties of Z[i]” on pages 6163.
Reading this section, taking notes, and studying the examples should take approximately 15 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Ring Properties of Z[i] Exercise”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Ring Properties of Z[i] Exercise” (PDF)
Instructions: Try to do Exercise 1 on page 63.
After attempting the exercise assigned above, discuss your solution in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 30 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Introduction to Section 3.3: Gaussian Integers and Section 3.3.1: Ring Properties of Z[i]”

4.2.2 Sage Lab: “Arithmetic of Gaussian Integers: Geometry and Algebra”
 Web Media: The Saylor Foundation’s Sage Lab: “Arithmetic of Gaussian Integers: Geometry and Algebra”
Link: The Saylor Foundation’s Sage Lab: “Arithmetic of Gaussian Integers: Geometry and Algebra” (PDF)
Instructions: Download the linked set of labs. Upload the fifth one (4.4.2SageWS5.sws) to the Sage website where you created an account (subunit 1.4.2). Work through the lab carefully.
Completing this assignment should take approximately 30 minutes.
 Web Media: The Saylor Foundation’s Sage Lab: “Arithmetic of Gaussian Integers: Geometry and Algebra”

4.2.3 Division of Gaussian Integers: A Precise Algorithm
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 3.3.2: Division”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 3.3.2: Division” (PDF)
Instructions: Read “Section 3.3.2: Division” on pages 63–68. It looks long, but has lots of pictures.
Reading this section, taking notes, and studying the examples should take approximately 25 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Gaussian Division Exercise”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Gaussian Division Exercise” (PDF)
Instructions: Try to do Exercise 1 on page 68.
After attempting the exercise assigned above, discuss your solution in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 30 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 3.3.2: Division”

4.2.4 Primality
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 3.3.3 Primality”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 3.3.3 Primality” (PDF)
Instructions: Read “Section 3.3.3: Primality” on pages 6869.
Reading this section, taking notes, and studying the examples should take approximately 10 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Gaussian Primality Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Gaussian Primality Exercises” (PDF)
Instructions: Try to do Exercises 13 and Exercise 6 on page 69.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 1 hour.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 3.3.3 Primality”
 4.3 Algebraic and Transcendental Numbers

4.3.1 “Ring Properties” of the Algebraic Numbers
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Introduction to Section 3.4: Algebraic and Transcendental Numbers and Section 3.4.1: The Algebraic Numbers form a Ring”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Introduction to Section 3.4: Algebraic and Transcendental Numbers and Section 3.4.1: The Algebraic Numbers form a Ring” (PDF)
Instructions: Read the introduction to “Section 3.4: Algebraic and Transcendental Numbers” and all of “Section 3.4.1: The Algebraic Numbers form a Ring” on pages 7072.
Reading this section, taking notes, and studying the examples should take approximately 15 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Algebraic Rings Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Algebraic Rings Exercises” (PDF)
Instructions: Try to do Exercises 13 on page 74.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 1 hour.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Introduction to Section 3.4: Algebraic and Transcendental Numbers and Section 3.4.1: The Algebraic Numbers form a Ring”

4.3.2 Liouville’s Number
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 3.4.2: Liouville’s Number”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 3.4.2: Liouville’s Number” (PDF)
Instructions: Read “Section 3.4.2: Liouville’s Number” on pages 7274.
Reading this section, taking notes, and studying the examples should take approximately 15 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Liouville’s Number Exercise”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Liouville’s Number Exercise” (PDF)
Instructions: Try to do Exercise 4 on page 75.
After attempting the exercise assigned above, discuss your solution in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 15 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 3.4.2: Liouville’s Number”

Unit 5: Continued Fractions
In this, the second of our “sightseeing” units, we visit continued fractions.Continued fractionsare a surprisingly ancient topic that consists of writing numbers as fractions within fractions and notsimplifying them! Continued fractions enjoy several properties that allow us to find concise representations of both rational numbers and even of many irrational numbers.
Unit 5 Learning Outcomes show close
 5.1 Elementary Properties of Continued Fractions

5.1.1 Rational Numbers have Finite Continued Fractions
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Introduction to Chapter 7 and Section 7.1: Basic Notations”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Introduction to Chapter 7 and Section 7.1: Basic Notations” (PDF)
Instructions: Read the beginning of “Chapter 7 Introduction to Continued Fractions” and all of “Section 7.1 Basic Notations” on pages 155159.
Reading these sections, taking notes, and studying the examples should take approximately 30 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory: Continued Fractions Basic Notation Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory: Continued Fractions Basic Notation Exercises” (PDF)
Instructions: Try to do Exercises 15 on pages 159160.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 1 hour.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Introduction to Chapter 7 and Section 7.1: Basic Notations”

5.1.2 Convergents
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 7.2: Main Technical Tool”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 7.2: Main Technical Tool” (PDF)
Instructions: Read “Section 7.2: Main Technical Tool” on pages 160164.
Reading this section, taking notes, and studying the examples should take approximately 30 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Main Technical Tool Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Main Technical Tool Exercises” (PDF)
Instructions: Try to do Exercises 14 on page 164.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 1 hour.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 7.2: Main Technical Tool”

5.1.3 Sage Lab: “Continued Fractions”
 Web Media: The Saylor Foundation’s Sage Lab: “Continued Fractions”
Link: The Saylor Foundation’s Sage Lab: “Continued Fractions” (Sage)
Instructions: Download the linked set of labs. Upload the sixth one (5.1.3SageWS6.sws) to the Sage website where you created an account (subunit 1.4.2). Work through the lab carefully.
Completing this assignment should take approximately 30 minutes.
 Web Media: The Saylor Foundation’s Sage Lab: “Continued Fractions”

Unit 6: Multiplicative Functions, with Applications
Our final topic consists of multiplicative functions. We begin, not with a general analysis of multiplicative functions, but with Euler’s φfunction, which measures how many positive integers are smaller than a given integer and relatively prime to it. We then consider the sumofdivisors function σ. In both cases, we build up a general method to compute the value of the function, based on how an integer factors into primes.
Unit 6 Learning Outcomes show close
As you will see, φand σshare a property that allows us to evaluate them at any integer by factoring it into primes, then evaluating the function for these primes, or their powers. These functions are easy to evaluate for prime numbers, which makes it easy to evaluate then for any integer. After discussing some consequences of this common property, we discuss two other multiplicative functions of interest.
We finally turn to some applications of multiplicative functions. One of them is a purely mathematical application: that of computing perfect numbers, which are the sum of their divisors. You should already see that the function σis of interest in this case. Perfect numbers are related to Mersenne primes, which turn out to be very difficult to come by. In an ironic way, they are even more difficult to come by than Mersenne himself envisioned! As we're already analyzing the mistaken claims of a great number of theorists  a phenomenon which should both inspire and intimidate you  we also discuss the Fermat numbers, all of which Fermat thought would be prime, but only few are.
After these diversions inspired by σ, we come in a very fitting way to φ, the function that began our investigations of multiplicative functions. We can use φ to generalize Fermat’s Little Theorem from Unit 3. Euler’s Theorem essentially shows, yet again, how we can generalize an idea about numbers that are prime to numbers that are relatively prime.
A moment ago, we said that these multiplicative functions are easy to compute once we have a factorization of an integer. Hopefully, you remember what we said in Unit 1 about factoring: it is a deceptively simple problem. How deceptive is its simplicity? A trio of mathematicians formulated an algorithm for private communication whose security is based entirely on the premise that an eavesdropper could understand the communication if she could but factor a large integer into two primes. That’s it! The RSA algorithm, named after its inventors, thus draws together the main strands of this course into a tourdeforce of elegant simplicity: prime numbers, the greatest common divisor, relatively prime numbers, congruence, and multiplicative functions.

6.1 Euler’s φFunction
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 4.2.2: Euler’s φFunction”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 4.2.2: Euler’s φFunction” (PDF)
Instructions: Read “Section 4.2.2: Euler’s φFunction” on page 86.
Reading this section, taking notes, and studying the examples should take approximately 5 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Euler’s φFunction Exercise”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Euler’s φFunction Exercise” (PDF)
Instructions: Try to do Exercise 3 on page 86.
After attempting the exercise assigned above, discuss your solution in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 1 hour.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 4.2.2: Euler’s φFunction”

6.1.1 Sage Lab: “Computing φ(n)”
 Web Media: The Saylor Foundation’s Sage Lab: “Computing φ(n)”
Link: The Saylor Foundation’s Sage Lab: “Computing φ(n)” (Sage)
Instructions: Download the linked set of labs. Upload the seventh one (6.1.1SageWS7.sws) to the Sage website where you created an account (Unit 1.4.2). Work through the lab carefully.
Completing this assignment should take approximately 30 minutes.
 Web Media: The Saylor Foundation’s Sage Lab: “Computing φ(n)”

6.1.2 Computing φ(n)
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 5.2.1: The Euler φFunction”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 5.2.1: The Euler φFunction” (PDF)
Instructions: Read “Section 5.2.1: The Euler φFunction” on pages 107110.
Reading this section, taking notes, and studying the examples should take approximately 30 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  The Euler φFunction Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  The Euler φFunction Exercises” (PDF)
Instructions: Try to do Exercises 1, 3, 5, 6 on pages 112113.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 1 hour.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 5.2.1: The Euler φFunction”
 6.2 The SumofDivisors Function, σ

6.2.1 Sage Lab: “Computing σ(n)”
 Web Media: The Saylor Foundation’s Sage Lab: “Computing σ(n)”
Link: The Saylor Foundation’s Sage Lab: “Computing σ(n)” (Sage)
Instructions: Download the linked set of labs. Upload the eight one (6.2.1SageWS8.sws) to the Sage website where you created an account (subunit 1.4.2). Work through the lab carefully.
Completing this assignment should take approximately 30 minutes.
 Web Media: The Saylor Foundation’s Sage Lab: “Computing σ(n)”

6.2.2 σ(p) When p is Prime
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 5.2.2: The SumofDivisors Function”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 5.2.2: The SumofDivisors Function” (PDF)
Instructions: Read “Section 5.2.2 The SumofDivisors Function” on pages 110111.
Reading this section, taking notes, and studying the examples should take approximately 15 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  SumofDivisors Function Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  SumofDivisors Function Exercises” (PDF)
Instructions: Try to do Exercises 7 and 8 on page 113. Compute the sum of positive divisors only, not the number of divisors; you'll do those later.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 1 hour.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 5.2.2: The SumofDivisors Function”
 6.3 The Common Thread: Multiplicative Functions

6.3.1 Properties of Multiplicative Functions
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Introduction to Chapter 5 and Section 5.1: Definitions and Properties”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Introduction to Chapter 5 and Section 5.1: Definitions and Properties” (PDF)
Instructions: Read the introduction to “Chapter 5 Multiplicative Number Theoretic Functions” and all of “Section 5.1: Definitions and Properties” on pages 103106.
Reading these sections, taking notes, and studying the examples should take approximately 30 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Multiplicative Functions Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Multiplicative Functions Exercises” (PDF)
Instructions: Try to do Exercises 1 and 2 on pages 106107.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 15 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Introduction to Chapter 5 and Section 5.1: Definitions and Properties”

6.3.2 Two Other Multiplicative Functions: τ, μ
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 5.2.3: The NumberofDivisors Function and Section 5.3: The Mobius Function and the Mobius Inversion Formula”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 5.2.3: The NumberofDivisors Function and Section 5.3: The Mobius Function and the Mobius Inversion Formula” (PDF)
Instructions: Read “Section 5.2.3: The NumberofDivisors Function” on pages 111112 and “Section 5.3: The Mobius Function and the Mobius Inversion Formula” on pages 113116.
Reading these sections, taking notes, and studying the examples should take approximately 30 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  NumberofDivisors and Mobius Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  NumberofDivisors and Mobius Exercises” (PDF)
Instructions: Try to do Exercises 7 and 8 on page 113. Compute the number of positive divisors only, not the sum of divisors, as you already computed them. Then Try to do Exercises 13 on page 116.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 1 hour.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 5.2.3: The NumberofDivisors Function and Section 5.3: The Mobius Function and the Mobius Inversion Formula”

6.4 Perfect Numbers
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Perfect, Mersenne, and Fermat Numbers, pages 116118”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Perfect, Mersenne, and Fermat Numbers, pages 116118” (PDF)
Instructions: Read the beginning of “Section 5.4: Perfect, Mersenne, and Fermat Numbers” on pages 116118 through Theorem 53 and its proof.
Reading this section, taking notes, and studying the examples should take approximately 15 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Perfect Numbers Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Perfect Numbers Exercises” (PDF)
Instructions: Try to do Exercises 1, 3, 4, and 5 on pages 120121.
Instructions: After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 15 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Perfect, Mersenne, and Fermat Numbers, pages 116118”
 6.5 Building Big Primes

6.5.1 Mersenne Primes and Fermat Numbers, or, How to Get Famous by Making Mistakes
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 5.4 Perfect, Mersenne, and Fermat Numbers”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 5.4 Perfect, Mersenne, and Fermat Numbers” (PDF)
Instructions: Finish reading “Section 5.4: Perfect, Mersenne, and Fermat Numbers” on pages 118120.
Reading this section, taking notes, and studying the examples should take approximately 30 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Mersenne Primes and Fermat Numbers Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Mersenne Primes and Fermat Numbers Exercises” (PDF)
Instructions: Try to do Exercises 7 and 9 on page 121.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 1 hour.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 5.4 Perfect, Mersenne, and Fermat Numbers”
 6.6 Euler’s Theorem

6.6.1 Sage Lab: Finding Multiplicative Inverses Modulo n
 Web Media: The Saylor Foundation’s Sage Lab: “Finding Multiplicative Inverses Modulo n”
Link: The Saylor Foundation’s Sage Lab: “Finding Multiplicative Inverses Modulo n” (Sage)
Instructions: Download the linked set of labs. Upload the ninth one (6.6.1SageWS9.sws) to the Sage website where you created an account (subunit 1.4.2). Work through the lab carefully.
Completing this assessment should take approximately 30 minutes.
 Web Media: The Saylor Foundation’s Sage Lab: “Finding Multiplicative Inverses Modulo n”

6.6.2 Euler’s Theorem
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 4.6: Theorems of Fermat, Euler, and Wilson”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 4.6: Theorems of Fermat, Euler, and Wilson” (PDF)
Instructions: Finish reading “Section 4.6: Theorems of Fermat, Euler, and Wilson” on pages 97100.
Reading this section, taking notes, and studying the examples should take approximately 30 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Theorems of Fermat, Euler, and Wilson Exercises”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Theorems of Fermat, Euler, and Wilson Exercises” (PDF)
Instructions: Try to do Exercises 1, 3, and 6 on page 101.
After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 30 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 4.6: Theorems of Fermat, Euler, and Wilson”
 6.7 RSA Encryption

6.7.1 Public Key Cryptography
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 9.1.1: Public Key Cryptography”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 9.1.1: Public Key Cryptography” (PDF)
Instructions: Read “Section 9.1.1: Public Key Cryptography” on pages 185186.
Reading this section, taking notes, and studying the examples should take approximately 15 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 9.1.1: Public Key Cryptography”

6.7.2 How the RSA Algorithm Works
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 9.1.2: The RSA Algorithm”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 9.1.2: The RSA Algorithm” (PDF)
Instructions: Read “Section 9.1.2: The RSA Algorithm” on pages 186189.
Reading this section, taking notes, and studying the examples should take approximately 20 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  RSA Algorithm Exercise”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  RSA Algorithm Exercise” (PDF)
Instructions: Try to do Exercise 1 on page 190.
After attempting the exercise assigned above, discuss your solution in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 1 hour.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 9.1.2: The RSA Algorithm”

6.7.3 Sage Lab: “Demonstration of RSA”
 Web Media: The Saylor Foundation’s Sage Lab: “Demonstration of RSA”
Link: The Saylor Foundation’s Sage Lab: “Demonstration of RSA” (PDF)
Instructions: Download the linked set of labs. Upload the last one (6.7.3SageWS10.sws) to the Sage website where you created an account (subunit 1.4.2). Work through the lab carefully.
Completing this assignment should take approximately 30 minutes.
 Web Media: The Saylor Foundation’s Sage Lab: “Demonstration of RSA”

6.7.4 Is the RSA Algorithm Safe?
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 9.1.3: Is RSA Safe?”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 9.1.3: Is RSA Safe?” (PDF)
Instructions: Read “Section 9.1.3: Is RSA Safe?” on pages 190191.
Reading this section, taking notes, and studying the examples should take approximately 10 minutes.  Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory  RSA Exercise”
Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory  RSA Exercise” (PDF)
Instructions: Try to do Exercise 1 on page 191. It’s not important to solve this problem; what’s important is to see how difficult it is!
After attempting the exercise assigned above, discuss your solution in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.
Completing this assessment should take approximately 30 minutes.
 Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory  Section 9.1.3: Is RSA Safe?”

Final Exam
 Final Exam: The Saylor Foundation’s “MA233 Final Exam”
Link: The Saylor Foundation’s “MA233 Final Exam”
Instructions: You must be logged into your Saylor Foundation account in order to access this exam. If you do not yet have an account, you will be able to create one, free of charge, after clicking the link.
 Final Exam: The Saylor Foundation’s “MA233 Final Exam”