Math 104B: Number Theory II (winter 2013)
This is the second course in the Number Theory sequence. It presumes familiarity with Math 104A (Number Theory I). It will focus on characters, primitive roots, continued fractions, diophantine equations, quadratic forms and quadratic reciprocity.Meeting times:
- Lecture (by Alina Bucur) MWF 9-9:50pm, AP&M B412.
- Discussion section (by Tristan Sandler) Th 3-4pm, AP&M 2402.
| Instructor: Alina Bucur
Office: AP&M 7131
Office hours: Tu 2-3pm
and by appointment
| TA: Tristan Sandler|
Office: AP&M 6351
Office hours: W 2-4pm
and by appointment
Required text: W. LeVeque Fundamentals of Number Theory
It is available at the UCSD bookstore and the usual online sources (like Amazon). It is the same text as last quarter, so used copies should be available also. It will be on reserve at the S&E library.
I might supplement it with notes as the term progresses.
Homework: Assigned weekly, due in section at the beginning of the discussion or in the relevant homework box located in the basement of AP&M 15MIN BEFORE the start of the discussion. Tristan will stop and collect it before going to class.
Quizzes: There will be one or two quizzes, during lecture, to be announced at least a week in advance. They will be based on homework.
Midterm exam: Friday, February 8, in class.
Final exam: Wednesday, March 20, 8-11am, location TBD. Please note that by signing up for this course, you are agreeing to sit for the final examination at this date and time. The exam will cover material from the whole course, but weighted towards material not included on the midterm.
Grading: 30% homework + quizzes, 30% midterm, 40% final exam.
No textbooks, calculators are allowed during exams, but you may bring one 8 1/2 by 11 page of notes (both sides).
No make-up exams will be given and no late homework will be accepted. Cheating on an exam results in 0 points for that exam, as well as further disciplinary action. Please read very carefully the following ACADEMIC INTEGRITY GUIDELINES.
- The discussion time is about to change. You will receive an email from the undergrad coordinator in the Department of Mathematics with the new time and location. Keep an eye out for it in your official ucsd email account. You should receive by the end of today, W 1/9.
Please note that in order for this change to take place, the class will have to be cancelled and readded into the system. Which means it will have a new section id from the Registrar. You will have to add the class again once the new id is in the system.
- Discussion has been rescheduled. See above. Please register for the class AGAIN!
- No class on Friday, January 18. Monday 1/21 is a holiday, so no class then either.
- Alina's office hours are cancelled for Tu 1/22. Instead, I will have office hours Th 1/24, 10:30-11:30am.
- The first quiz will be Friday, February 1, in class. Please be on time, as we will start the quiz promptly at 9am and you will have only 20min.
- Special office hours midterm week: Th 2/7 5-6pm (Alina)
- The second quiz will be Wednesday, March 6, in class. Please be on time, as we will start the quiz promptly at 9am and you will have only 20min.
- Alina's office hours on Tu 3/12 are cancelled. Please email if you have any questions or if you need to schedule an appointment.
- Office hours for the exam week: Tu 3/19 2-4pm AP&M B412.
- HW 1 (due Th 1/17) from the textbook
Hint for problem 9: follow the hint in the book and then use the fact that if a,b are two relatively prime integers and their product ab is a sum of two squares, then both a and b have to be sums of two squares.
- Section 5.1: 2,4
- Section 5.2: 2,3,6,8,9
- HW 2 (due Th 1/24) pdf
- HW 3 (due Th 1/31) pdf
- HW 4 (due Th 2/7) pdf
- HW 5 (due Th 2/14) pdf
- HW 6 (due Th 2/21) pdf
- HW 7 (due Th 2/28) pdf
- HW 8 (due Th 3/7) pdf
- HW 9 (due Th 3/14) from the textbook
- Section 6.2: 3, 7, 10a
- Section 6.4: 2, 4
- Section 6.5: 1
- Lecture 1 (M 1/7) Review
- Lecture 2 (W 1/9) Defined Legendre symbol modulo an odd prime, proved Thm 3.22, 3.23, 5.3 and 5.5 from the book.
- Lecture 3 (F 1/11) Gauss's lemma and applications
- Lecture 4 (M 1/14) The law of quadratic reciprocity
- Lecture 5 (W 1/16) Congruence conditions for quadratic residues; Theorem 5.8
- Lecture 6 (W 1/23) Finished the proof of Theorem 5.8. Definition and properties of the Jacobi symbol.
- Lecture 7 (F 1/25) More about Jacobi symbols. Here we diverge from the textbook a bit, so I wrote up notes for this part of the course and posted them below. In class we covered up to the middle of the proof of Prop 4.
- Lecture 8 (M 1/28) Fisnished Prop 4 and started Thm 5.
- Lecture 9 (W 1/30) Finished the proof of Theorem 5. Pythagorean triples: definition.
- Lecture 10 (F 2/1) Finished Pythagorean triples. Started looking at Fermat's equation with n=4.
- Lecture 11 (M 2/4) Fermat's equation with n=4; units in Z[i]; which primes from Z remain prime in Z[i]?
- Lecture 12 (W 2/6) More about primes and gaussian integers: Euler's strategy. We proved the reciprocity step and started working on the descent step.
- Midterm (F 2/8) no lecture
- Lecture 13 (M 2/11) Discussed the midterm and finished the descent argument for primes in the gaussian integers.
- Lecture 14 (W 2/13) Divisibility and the norm map for gaussian integers.
- Lecture 15 (F 2/15) Unique factorization, euclidean structure for gaussian intgers. Relatively prime gaussian integers. Applications to the arithmetic of Z: Fermat numbers, Pytgorean triples revisited.
- Lecture 16 (W 2/20) Applications of gaussian integers to diophantine equations.
- Lecture 17 (F 2/22) More applications of gaussian integers to diophantine equations. Prime elements in the ring of gaussian integers.
- Lecture 18 (M 2/25) Integers as sums of two squares.
- Lecture 19 (W 2/27) Which numbers can be written as sum of two squares? Section 6.1: multiplicative and completely multiplicative functions, divisor functions.
- Lecture 20 (F 3/1) Perfect numbers, Riemann zeta function, problem 5, section 6.1.
- Lecture 21 (M 3/4) The Möbius function and Möbius inversion.
- Lecture 22 (W 3/6) The big O, little o, ∼, ≪ notations (Section 6.4)
- Lecture 23 (F 3/8) The Euler constant, rate of convergence, Euler-Maclaurin summation.
- Lecture 24 (M 3/11) Finished Euler-Maclaurin summation. Applications.
- Lecture 25 (W 3/13) Eratosthenes and Brun's sieve (Section 6.5).
- Lecture 26 (F 3/15) Review