Math 115 - Introduction to Number Theory

InstructorPaul Vojta
LecturesTuTh 12:30–2:00, Hearst Mining 310
Course Control Number18246
Office883 Evans
E-Mailvojta@math.berkeley.edu
Office HoursTuesdays 11–12, Wednesdays 1–2, Thursdays 11–12.
Please email me to set up a time to meet if you cannot make any of these times.
PrerequisitesMath 53 and 54
Required TextNiven, Zuckerman, Montgomery, An introduction to the theory of numbers, Wiley, 5th edition

Here are some links that you may find useful if you do not have the most recent printing of the textbook.

Syllabus As currently planned, the following parts of the book will be covered:

Chapter 1 - Divisibility
All sections

Chapter 2 - Congruences
Through the first two paragraphs of Section 2.9.

Chapter 3 - Quadratic Reciprocity and Quadratic Forms
All except Section 3.6 and pages 174–175.

Chapter 4 - Some Functions of Number Theory
Sections 4.1 through 4.3.

Chapter 7 - Simple Continued Fractions
Sections 7.1 through 7.8.

Plans are subject to change.

GradingGrading will be based on:
30%Homework Assigned weekly
15%First midterm Thursday, February 1612:30–2:00 pm
20%Second midterm Thursday, March 2312:30–2:00 pm
35%Final examThursday, May 11 3–6 pm

The final exam will be held in our usual classroom (Hearst Mining 310).

HomeworkAssigned weekly, generally due on Thursdays. Assignments are given below. Solutions will be posted on bCourses.
CommentsNone yet. Watch this space for further developments.

Homework Assignments

As a general rule, for any assigned problem from the book, you may use without proof any earlier exercise in the book. Starting with the second assignment, when doing a problem from a given section of the book, you may not use material from subsequent sections of the book (even if problems from those sections are also in the homework assignment).

DueSectionProblemsComments
1/26 1.21c, 2, 3e, 14, 19 (n>=2), 21 For 3e, use a method that would be practical with much larger numbers in place of 6, 10, and 15.
2/2 1.36, 14, 16, 19, 20, 21
1.42, 4, 6, 9
2.12, 6, 8
2/9 2.112, 18, 20, 27, 40, 44, 59 For #18, p is prime.
Tuesday 2/14 2.26, 8
2.34, 7, 14, 16, 17, 20 For #4, use a method that would work with much larger numbers in place of 1, 2, 3 and 3, 4, 5. Exception: When using the Euclidean algorithm, you may omit details, but do say “Euclidean Algorithm”.
2/23 2.46, 9, 10
2.51, 2, 4 For #4 (and for other exercises in the book), the notation (H) means that there is a hint in the back of the book (pages 503–511).
2.61, 2, 5, 8, 9, 10
3/2 2.71a, 2, 6, 7
2.86, 8, 10
3/9 2.811, 13, 16, 19, 20, 21 For #13, p is a prime.
For #16, let p be a prime factor of the gcd, and argue by contradiction.
2.91a, 7
3/16 3.13, 7, 11, 12, 15 Don't use methods from Section 3.2 or 3.3.
3.26, 7, 8, 18 Don't use methods from Section 3.3.
For #18, if you have a very old version of the textbook, replace 1111111111111 with 1111118111111 (twice). (1111111111111 is not prime.)
3.31, 2, 4
Tuesday 3/21 3.36, 7, 8, 15, 21
3.41, 3, 8, 10
4/6 3.51, 3, 5, 6, 11, 12 For Exercise 3.5.5, the first sentence should read: “Show that x2+5y2 and 2x2+2xy+3y2 are the only reduced positive definite quadratic forms of discriminant -20.”
3.71, 5
4/13 4.12, 3, 4, 6, 9
4.25, 6, 10, 12, 16
4.31, 2, 5, 8
4/20 7.11, 4, 5
7.33abd, 5
7.42
4/27 Download:    pdf    dvi Correction: The last problem needs to require c>0.
For Exercise 7.5.3, if you are using an old version of the textbook, see the corrections in the links at the top of this page.
Hints for the last problem: Try “⇐” first. Also, the lemma from Tuesday's class may be helpful for “⇒”.
Do not
hand in
Download:    pdf    dvi

Course Handouts

DateTitleDownload
April 4 Proof of the existence of the greatest-integer function pdf dvi
April 6 Why quadratic forms are important in number theory pdf dvi
April 27 A different proof of Theorem 7.26 pdf dvi

Exams (Generally)

Policies for exams are as follows.

The two midterms will be given during the normal class hours (12:30–2 pm), and will be in our normal classroom (Hearst Mining 310).

Generally, about a week before each exam, a sample exam will be distributed in class and posted on bCourses. This will usually be an exam from an earlier Math 115 class that I've taught. Sample exams should be used to get a general idea of the likely length of an exam and the general nature of questions to be asked (e.g., the balance between computational and more theoretical questions). However, one should not (for example) note that a sample exam contains questions on material from Sections 1.5, 2.1, 2.7, 3.1, 3.4, etc., and expect to see questions from those sections on the actual exam.

Exams are cumulative, so the second midterm may have questions from material prior to the first midterm. Of course, the final exam will cover the whole course, but will have increased emphasis on the material not covered on the midterms.

Here is a link "How to lose marks on math exams" (by a former GSI Andrew Critch).

The Math Department maintains an archive of old exams (usually without answers). Here is the link for Math 115.

And finally, a word about regrades: Grade calculation errors are welcome for discussion or review. Whether this solution should be worth 4 or 5 points is not.

First Midterm

The first midterm was given on Thursday, February 16, from 12:40 to 2:00, in our usual classroom. It covered everything in the textbook up to but not including Theorem 2.19 (page 69). As exceptions, the content on the Prime Number Theorem were not required, nor was the example of lack of prime factorization discussed starting on the bottom of page 21 and continuing through the middle of page 23.

A sample midterm was distributed in class on February 9, and is available on bCourses. Solutions to the sample midterm are also available now on bCourses.

Solutions to the midterm itself are also now available on bCourses.

The (very rough) curve for the midterm is:

A66–74
B44–65
C28–43

The median was 54, the mean was 53.6, and the standard deviation was 15.5.

Second Midterm

The second midterm will be given on Thursday, March 23, from 12:40 to 2:00, in our usual classroom. It will cover everything in the textbook up to and including Section 3.4; lectures up to and including Tuesday, March 14; and homeworks up to and including the homework due March 21 (with the same exceptions as for the first midterm). It will be cumulative, but material since the first midterm will be emphasized.

A sample midterm was distributed in class on March 16, and is available on bCourses. Solutions to the sample midterm are also available now on bCourses.

Solutions to the midterm itself are also now available on bCourses.

The (very rough) curve for the midterm is:

A76–97
B55–75
C37–54

The median was 65, the mean was 65.7, and the standard deviation was 19.0.

RRR Week and Beyond

On May 2 (Tuesday of RRR week) we met and reviewed the material in Chapters 1–3. On Thursday, we will review material from Chapters 4 and 7.

A sample final exam was distributed in class on May 2, and is also available on bCourses. Solutions are also available now on bCourses.

Solutions for the final exam are now available on bCourses.

The median was 105, the mean was 108.1, and the standard deviation was 35.0. A curve will be posted shortly.


Last updated 20 May 2017