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

**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 10.22

Schedule -- updated 10.22

- 10.10: Homework 6 is due on
**Friday, Oct. 17**(and is very short). Homework 7 will be posted on Wednesday, Oct. 15 and is due the following Wednesday. - 10.1: Information about Midterm 1: content, format, and a list of 8 statements which you may be asked to prove.
- 9.12: Zoe Hamel, our TA, will be in the Math Learning Centre 5pm-7pm on Tuesdays and 2:30-4:30pm on Wednesdays.
- 9.4: The content of the 5th edition of the textbook is similar enough that it should be OK, but it's your responsibility to make sure that you are reading the correct sections.
- 9.3: If you need help with registration, please see this page for instructions.
- 9.3: If you need to arrange accommodations for a disability/difference (e.g., taking exams at the Access & Diversity Center), please let me know as soon as possible.
- 9.3: The midterm exam dates for this course are
**Thurs, Oct. 9**and**Thurs, Nov. 6**, both exams 6:30--7:30pm in 202 Macleod. Please double-check these dates against the exam dates for your other courses and let me know ASAP if you have a conflicting exam.

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 | Homework 8 | ||

9 | 11.5 | |||

10 | 11.12 | |||

11 | 11.19 | |||

12 | 11.26 |

''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 |
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1 | 9.3 | Course overview & policies; survey | Course syllabus Abbreviations |
Survey | ||

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). |
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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! |
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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.) |
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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) |
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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 examplesDivision with congruences: §4.1 Thm. 4.5Solvability of linear congruences: §4.2 intro + Thm. 4.11 |
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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. |
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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 | Hensel's Lemma: statement, proof sketch, and examples; recursive formula in Case 1 | §4.4 from Thm. 4.15-end. (Focus on the examples rather than the proof of HL. The statement of Cor. 4.15.1 is useful: it is a recursive formula for the unique lifts of a Case 1 solution to the congruence mod p.) | ||||

10.17 | More examples of lifting polynomial congruences; Wilson's theorem |
Wilson's Theorem: §6.1 through the proof of Thm. 6.1. |
HW 6 due | |||

8 | 10.20 | Converse of Wilson's theorem; Fermat's Little Theorem; pseudoprimes and Carmichael numbers | Converse of Wilson's theorem: §6.1, Thm. 6.2 and Example 6.3.Fermat's Little Theorem and applications: §6.1, Thm. 6.3 through Cor. 6.5.1. Pseudoprimes and Carmichael numbers: §6.2 through Example 6.12. (No need to spend a lot of time on this section now; we will come back to it.) |
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10.22 | Euler's theorem; calculating values of Euler's φ-function |
§6.3 (just the statement of Euler's Theorem and the examples) | HW 7 due | |||

10.24 | Proof of Euler's Theorem; examples and applications | Proof of Euler's Theorem: §6.3, proofs of Thm. 6.13 and 6.18. |
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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 |
§2.2 | ||||

11 | 11.10 | Primality testing, part 1: using FLT and Euler's criterion |
§6.2 §11.4 |
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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 |
§4.6 §6.1: p. 221--end |
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12 | 11.17 | Factorization techniques, part 2: state of the art |
TBA | |||

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 |
§8.4 | ||||

13 | 11.24 | Public key crypto, part 3: Diffie-Hellman key exchange |
§8.6 | |||

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 | ||||

TBA | Review session | |||||

TBA | Final exam | Cumulative |