Math 312 • Introduction to Number Theory • UBC, Fall 2014

the Euler phi function

Lectures: MWF 11:00-12:00 in LSK 460
Office hours: MW 9:50-10:50 and by appointment, LSK 300C

Instructor: Laura Peskin, lpeskin at math dot ubc dot ca
Grader: Zoe Hamel, zhamel at math dot ubc dot ca

Textbook: Rosen, Elementary Number Theory & Its Applications, 6th ed.

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Syllabus and Course Policies -- updated 10.1
Announcements -- updated 10.10
Assignments --updated 10.16
Schedule -- updated 10.10


Syllabus and Course Policies

This course is an introduction to the multiplicative structure of the integers. We will cover classical theorems such as the Fundamental Theorem of Arithmetic, the Chinese Remainder Theorem, and the Law of Quadratic Reciprocity, as well as a variety of modern applications. At the same time, we'll build skill in mathematical reasoning and proof-writing. For a detailed list of topics, see the
course syllabus or the day-by-day schedule.

Grades will be calculated using the following formula: 20% weekly homeworks, 30% midterm exams (two exams, weighted equally), 50% final exam. The two lowest homework scores will be dropped. See the course syllabus for policies on collaboration, exams, and late work/absences.


Assignments are due on Wednesdays at 11am in class, except for HW 6 which is due on Friday, Oct. 17. No late homework will be accepted. The lowest two scores will be dropped.

# Due date Assignment Grading scheme Solutions
1 9.10 Homework 1 5 pts each for #2, #4, #10
5 pts for completeness
Solutions to HW 1
2 9.17 Homework 2 5 pts each for #3, #5, #7
5 pts for completeness
Solutions to HW 2
3 9.24 Homework 3 9 pts for #5, 6 pts for #7
5 pts for completeness
Solutions to HW 3
4 10.1 Homework 4   Solutions to HW 4
5 10.8 Homework 5   Solutions to HW 5
6 10.17 (!!) Homework 6    
7 10.22 Homework 7    
8 10.29      
9 11.5      
10 11.12      
11 11.19      
12 11.26      

Course Schedule

All chapter/section references are to the textbook Elementary Number Theory and Its Applications (6th ed).
''Through'' means ''up to and including''; for example ''§3.1 through Example 3.3'' means ''§3.1 up to the end of Example 3.3.''

Week Date Topics Required reading In-class exercise Optional reading Comments
1 9.3 Course overview & policies; survey Course syllabus
9.5 Review of induction;
Division algorithm
Induction: §1.3, focusing on the first principle of induction.
Extra example: proving the Binomial Theorem by induction.

Divisibility: §1.5. (We didn't get to the final part of §1.5, "Greatest common divisors," and will begin there on Monday.)
  To see a true "division algorithm" which produces q and r given a dividend a and divisor b, look here.

Both the principle of induction and the division algorithm can be justified using the Well-Ordering Property of the natural numbers. Read p.6 of the text, then proofs of Thm 1.5 (p.25) and of Thm. 1.10 (p.37).
2 9.8 Greatest common divisor;
Prime numbers: definition, Euclid's
proof of infinitude, sieving
GCD: last page of §1.5.

Primes and sieves: §3.1 through Example 3.3
Sieve of Eratosthenes

Graph of π(x) with scaled and unscaled axes
Here is an animation of the Sieve of Eratosthenes on the integers up to 121. The sieve "meshes" are 2, 3, 5, 7, and 11 = sqrt(121).
9.10 Finish up Sieve of Eratosthenes; Properties of gcd: Bezout's theorem,
linear combinations.
§3.3 through Theorem 3.9. (The main topics of lecture were Thm. 3.8 and Thm. 3.9.)     HW 1 due
9.12 Euclidean algorithm
(calculating gcd, part 1),
Solving linear equations with the Extended Euclidean algorithm
§3.4, also see "in-class exercise" notes for several examples Euclidean algorithm    
3 9.15 Linear Diophantine equations:
no solution vs. infinitely many
§3.7 through Example 3.29. The main topic was Thm. 3.23. To prove it, we needed (and proved) Lemma 3.4 (p.113) and Thm. 3.6 (p.94).

Also see the in-class exercise from 9.17 for examples.
Here is a scan of a 1910 English translation by Thomas Heath of Diophantus's Arithmetica. (With commentary; text begins on p.129.) Offered mainly for historical value; it's pretty hard to read!
9.17 gcd of > 2 numbers;
linear Diophantine equations in > 2 variables
gcd of > 2 numbers: §3.3, p.98-end

Linear equations in > 2 variables: §3.7, Thm. 3.24-end
Solving linear Diophantine equations in 2 variables We have an algorithmic way to tell whether any linear Diophantine equation has an integer solution: just find the gcd of the coefficients (e.g. using Euclid's algorithm) and check if this gcd divides the constant term.

But it's impossible to construct an algorithm to decide whether a general Diophantine equation has an integer solution: this impossibility was proved by Matiyasevich in 1970, answering a question posed by Hilbert in 1900. Here (log in through UBC library) is a paper by Martin Davis explaining the proof.
HW 2 due
9.19 Finish up Diophantine equations;
Fundamental Theorem of Arithmetic
Diophantine equations: Finish proof of Thm. 3.24 in §3.7

Fundamental Thm of Arithmetic: §3.5 through Lemma 3.7. (We'll finish the uniqueness part of the FTA on Monday. We won't do Lemma 3.7 in class but please read it.)
4 9.22 Uniqueness part of the Fund. Thm. of Arith;
Congruences: definition, existence, and basic properties
Proof of uniqueness part of FTA: Thm. 3.15 in §3.5.

Congruences: §4.1 Defn. of congruence modulo m (p. 145), defn. of complete system of representatives/residues mod m (p. 148)
9.24 Arithmetic with congruences;
Representing integers with respect to a base;
modular exponentiation
Arithmetic with congruences: §4.1 Thms. 4.2, 4.4, and 4.8.

Representations of integers: §2.1 (though we used a different method for finding binary representations; see "in-class exercise").
Binary and hexadecimal   HW 3 due
9.26 Arithmetic with congruences §4.1 Thms. 4.4, 4.6, and 4.8. Finish binary/hex exercise (posted above)    
5 9.29 Modular exponentiation; division with congruences; solvability criterion for linear congruences Modular exponentiation: final section of §4.1 (omit Thm. 4.10), and two more examples

Division with congruences: §4.1 Thm. 4.5

Solvability of linear congruences: §4.2 intro + Thm. 4.11
10.1 Proof of solvability criterion for linear congruences (Thm. 4.11); modular inverses §4.2 Solving linear congruences: Problems, Solutions   HW 4 due
10.3 Solving systems of congruences, part 1: Chinese Remainder Theorem § 4.3 through Example 4.16; proof of Thm. 4.13 will be done on Monday. You just need to know the process for the exam. Look at Example 4.17 for another way of solving systems of linear congruences (probably similar to the method you used to solve systems of linear equations in high school).

The Chinese Remainder Theorem is going to be important when we learn about the RSA cryptosytem (p. 323-329 in the text). In particular, we'll look at systems of two congruences, where the two moduli are distinct primes.
6 10.6 Proof of uniqueness (mod M) in the Chinese Remainder Theorem §4.3 proof of Thm. 4.13    
10.8 CRT with non-relatively prime moduli; Review Q & A; Prime Number Theorem Notes on CRT not assuming moduli are relatively prime   HW 5 due
10.9 Midterm Exam 1 Material of Sept 5 through Oct 3 6:30-7:30pm 202 Macleod
10.10 Polynomial congruences §4.4 through case (1) of Hensel's Lemma      
7 10.13 No lecture (Thanksgiving)        
10.15 Quadratic residues, part 1:
Euler's criterion, Gauss's lemma
§11.1 (omit Thm. 11.2)    
10.17 Quadratic residues, part 2:
Quadratic reciprocity
§11.2     HW 6 due
8 10.20 Quadratic residues, part 3:
Jacobi symbols
§ 11.3      
10.22 Euler's theorem;
multiplicativity of Euler's φ-function
§7.1 through Thm. 7.4
    HW 7 due
10.24 Calculating φ(n);
definition of multiplicative order
§ 7.1
§9.1 through Example 9.1
9 10.27 Primitive roots: definition,
examples, nonexamples
§9.1: Thm. 9.1--end      
10.29 Primitive roots modulo primes:
Lagrange's theorem
§9.2     HW 8 due
10.31 Existence criterion for primitive roots §9.3      
10 11.3 Discrete logarithms and index arithmetic §9.4      
11.5 Review day:
Reciprocity revisited
TBA     HW 9 due
11.6 Midterm Exam 2 Material of Oct 10 through Nov 3 6:30-7:30pm 202 Macleod
11.7 Toolkit for applied number theory:
algorithmic efficiency, benchmarks
11 11.10 Primality testing, part 1:
using FLT and Euler's criterion
11.12 Primality testing, part 2:
using orders and primitive roots
§9.5     HW 10 due
11.14 Factorization techniques, part 1:
Pollard Rho, Pollard p - 1
§6.1: p. 221--end
12 11.17 Factorization techniques, part 2:
state of the art
11.19 Public key crypto, part 1:
protocols, textbook RSA
§8.4   HW 11 due
11.21 Public key crypto, part 2:
attacks on RSA
13 11.24 Public key crypto, part 3:
Diffie-Hellman key exchange
11.26 Public key crypto, part 4:
Hash functions, digital signatures
TBA     HW 12 due
11.28 Choice of: pseudorandom numbers,
ElGamal, elliptic curve crypto
  TBA Review session      
TBA Final exam Cumulative