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 φ-function, which measures how many positive integers are smaller than a given integer and relatively prime to it. We then consider the sum-of-divisors 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.
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 tour-de-force of elegant simplicity: prime numbers, the greatest common divisor, relatively prime numbers, congruence, and multiplicative functions.
Unit6 Learning Outcomes
Upon successful completion of this unit, you will be able to:
- identify the criteria of a multiplicative function, and compute the
values of such functions for integers that are “easy” to factor;
- identify important multiplicative functions, especially Euler’s
φ-function, the sum-of-divisors function σ;
- describe the relationships between perfect numbers, the σ-function,
and Mersenne primes;
- search for large prime numbers using ideas of Mersenne and Fermat;
- describe how Euler’s φ-function allows us to generalize Fermat’s
Little Theorem; and
- describe the mathematical theory of the RSA encryption scheme, and
successfully encrypt and decrypt short messages using simple
parameters.
6.1 Euler’s φ-Function - Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Section 4.2.2: Euler’s φ-Function” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Section 4.2.2: Euler’s φ-Function” (PDF)
Instructions: Read “Section 4.2.2: Euler’s φ-Function” 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 φ-Function Exercise” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Euler’s φ-Function 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.
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 (6.1.1-SageWS7.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.
6.1.2 Computing φ(n) - Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Section 5.2.1: The Euler φ-Function” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Section 5.2.1: The Euler φ-Function” (PDF)
Instructions: Read “Section 5.2.1: The Euler φ-Function” on pages
107-110.
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 φ-Function Exercises” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - The Euler φ-Function Exercises” (PDF)
Instructions: Try to do Exercises 1, 3, 5, 6 on pages 112-113.
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.
6.2 The Sum-of-Divisors 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
(6.2.1-SageWS8.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.
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 Sum-of-Divisors Function” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Section 5.2.2: The Sum-of-Divisors Function” (PDF)
Instructions: Read “Section 5.2.2 The Sum-of-Divisors Function” on
pages 110-111.
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 - Sum-of-Divisors Function Exercises” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Sum-of-Divisors 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.
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 103-106.
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 106-107.
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.
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 Number-of-Divisors 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 Number-of-Divisors Function and Section 5.3: The Mobius Function and the Mobius Inversion Formula” (PDF)
Instructions: Read “Section 5.2.3: The Number-of-Divisors Function”
on pages 111-112 and “Section 5.3: The Mobius Function and the
Mobius Inversion Formula” on pages 113-116.
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 - Number-of-Divisors and Mobius Exercises” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Number-of-Divisors 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 1-3 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.
6.4 Perfect Numbers - Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Perfect, Mersenne, and Fermat Numbers, pages 116-118” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Perfect, Mersenne, and Fermat Numbers, pages 116-118” (PDF)
Instructions: Read the beginning of “Section 5.4: Perfect,
Mersenne, and Fermat Numbers” on pages 116-118 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 120-121.
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.
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 118-120.
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.
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
(6.6.1-SageWS9.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.
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 97-100.
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.
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 185-186.
Reading this section, taking notes, and studying the examples
should take approximately 15 minutes.
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
186-189.
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.
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
(6.7.3-SageWS10.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.
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
190-191.
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.