Math 115 - Introduction to Number Theory

InstructorPaul Vojta
LecturesMWF 11:10–12:00, Physics Building 385
Class Number 25242
Office883 Evans
Office Hours MWF 12:10–1:00 and MWF 3:10–4:00, excluding University holidays.
During Finals Week: By appointment only.
PrerequisitesMath 55 is recommended
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.

Catalog Description Divisibility, congruences, numerical functions, theory of primes. Topics selected: Diophantine analysis, continued fractions, partitions, quadratic fields, asymptotic distributions, additive problems.
Syllabus The following parts of the book were 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 7.1–7.5 and 7.7–7.8.

GradingGrading will be based on:
30%Homework Assigned weekly
15%First midterm Friday, October 111:10–12:00 PT
20%Second midterm Wednesday, November 311:10–12:00 PT
35%Final exam Monday, December 13 11:30 am–2:30 pm PT

All exams (including the final) will be held in our regular classroom (Physics Building 385) unless otherwise announced.

HomeworkAssigned weekly, generally due on bCourses at 11:59 pm on Mondays. Assignments are given below. Solutions will be posted on bCourses.
Resiliency Plans If necessary (e.g., because of pandemic conditions or poor air quality), we may switch to holding class via Zoom. If this becomes necessary, details (including the meeting link) will be announced on bCourses.

Homework Assignments

General rules on homework assignments are:

Homework assignments are due on bCourses on the days indicated below. The lowest three homework scores will be dropped (note that homework scores include two additional scores from clickers—see the section on Clickers, below).

No. Due Section Problems Comments
19/3 1.21c, 2, 3d,e, 14, 19, 21, 26 For 3d and 3e, use a method that would be still be practical if the coefficients were replaced by much larger numbers.
For 19, assume that the set is finite and has at least two elements.
29/13 1.243 Remember not to use prime factorizations when solving this one.
You may assume that a, b, and c are nonzero.
1.36, 14, 16, 19, 20 The book often writes (b,c) in place of gcd(b,c).
1.42, 4, 6, 9
39/20 2.12, 6, 8, 12, 18, 20, 27, 40, 44, 51, 59 In 18, p is prime.
2.26, 8, 9, 14 Exercise 14 has a hint on page 504 (this is what the notation “(H)” means in the statement of the exercise).
Also, do not use de Polignac's formula (Theorem 4.2) to do Exercise 14. That formula occurs on page 182, and we haven't covered it yet.
49/27 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. Also, when using the Euclidean algorithm, you may just say “Euclidean Algorithm” and leave out the details.
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, 3 For #3, use Hensel's lemma whenever possible or show that it would not help.
510/4 Download:    dvi    pdf
610/11 2.82, 3, 6, 8, 9, 10, 11, 13
710/18 Download:    dvi    pdf
810/25 3.112, 14, 15 Remember not to use methods from Sections 3.2 or 3.3.
3.26, 7, 8, 18 Remember not to use methods from Section 3.3.
In #18, replace 1111111111111 with 1111118111111 (1111111111111 is not prime). (This change has already been made in recent printings of the textbook.)
3.31, 2, 4, 6, 7, 8, 15, 21
911/1 3.41, 3, 8, 10
3.51, 3, 5, 6, 11, 12 The first sentence of #5 should read, “Show that ... are the only reduced positive definite ...”
1011/8 3.71, 2, 3 For #3, do only the first two sentences. In other words, you don't need to compute rf(n).
Feel free to look at Section 3.6 for ideas.
1111/15 3.75
4.12, 3, 4, 6, 9, 16
4.25, 6, 11, 12, 16
1211/22 4.31, 2, 5, 8
7.11, 4, 5
7.33abd, 5
1312/2 Download:    dvi    pdf As announced in class, this assignment was revised on December 1. Some problems were removed.
14 Do not
hand in
Download:    dvi    pdf Solutions are now posted on bCourses.


We will use iClickers in class. These must be the physical iClickers, not the iClicker app. The latter is not allowed, because use of cell phones during class can be very distracting.

iClickers can be obtained from the bookstore, or you may be able to buy a used one from another student.

The grades from iClicker use will be incorporated into the homework portion of the course grade.

Clicker grades will be determined as follows. For each class day (other than the first two), you will receive one clicker point for participation if you have provided answers (correct or not) to at least 50% of the clicker questions on that day, and one clicker point for correctness if you answer at least one clicker question correctly on that day. These will be reported to bCourses separately, but will be combined for the purposes of the next steps.

I will then drop the lowest four sessions (class days), and then combine that information with the homework scores by creating two ``fake'' assignment scores based on the total clicker scores.

The clicker points are not meant to be a big challenge. The vast majority of you should be receiving two clicker points on every day that you attend. The point of clickers is that you should be checking your own understanding of the material (including the reading) as the course progresses.

The clicker grading plan is subject to change.

Course Handouts

No.DateTitle Download
1August 27 Proof of the Existence of the Greatest-Integer Function
(revised August 28 to reflect the proof given in class)
dvi pdf
2August 27 Equivalence of three types of induction dvi pdf
3September 1 Slides from the lecture of September 1 dvi pdf
4September 20 Slides from the lecture of September 20 dvi pdf
5September 22 More complicated congruences dvi pdf
6September 22 Slides from the lecture of September 22 dvi pdf
7September 24 The stronger general Hensel's lemma dvi pdf
8September 29 Slides from the lecture of September 29 dvi pdf
9October 22 A Rephrased Theorem 3.10 dvi pdf
10October 25 Basic information on matrices dvi pdf
11November 1 Slides from the lecture of November 1 dvi pdf
November 12 Slides from the lecture of November 12 dvi pdf
November 17 Slides from the lecture of November 17 dvi pdf
November 19 Slides from the lecture of November 19 dvi pdf
November 22 Slides from the lecture of November 22 dvi pdf
12November 15 Why quadratic forms are important in number theory dvi pdf
November 29 Slides from the lecture of November 29 dvi pdf
December 1 Slides from the lecture of December 1 dvi pdf
13December 3 A Different Proof of Theorem 7.26 dvi pdf
14December 3 Solving the Pell Equation, by H. W. Lenstra pdf
December 3 Slides from the lecture of December 3 dvi pdf

Exams (Generally)

Policies for exams are as follows.

One more thing. If you are computing gcd(a,b) or finding x and y such that gcd(a,b)=xa+yb, and if |a|<20 and |b|<20, then you can use any method you want and don't have to include the details. If |a| or |b| is at least 20, though, you are expected to use the Euclidean Algorithm and to show your work.

The two midterms will be given during the normal class hours (11:10–12:00), and will be in our normal classroom (Physics 385).

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) observe 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 Friday, October 1, from 11:10 to 12:00 Pacific Time, in our usual classroom (Physics 385).

It covered:

We skipped, and therefore the exam did not cover:

As noted under the heading “Exams (Generally),” this was a fully closed-book exam.

A sample midterm was distributed in class on September 24, and is available on bCourses. Solutions to the sample midterm have also been posted on bCourses.

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

Second Midterm

The second midterm was given on Wednesday, November 3, from 11:10 to 12:00 Pacific Time, in our usual classroom (Physics 385).

It covered:

We skipped, and therefore the exam did not cover:

As noted under the heading “Exams (Generally),” this was a fully closed-book exam.

A sample midterm was distributed in class on October 27, and is available on bCourses. Solutions to the sample midterm are also now available on bCourses.

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

RRR Week

We will meet on Monday, Wednesday, and Friday of RRR week (at the usual time and place). We will review the course. We will not use clickers during RRR week.

Office hours will continue on their usual schedule during RRR week.

Final Exam

As noted above, the final exam was given on Monday, December 13, 11:30 am – 2:30 pm, in the regular classroom (Physics Building 385).

The exam started at 11:30, not 11:40 (so, not “Berkeley time”).

It covered the whole course (all of the reading, homework problems, and lectures). The list of sections covered is listed in “Syllabus” above.

We skipped, and therefore the exam did not cover:

Approximately 65–75% of the exam was on new material not covered on either of the midterms.

A sample final exam was distributed in class on December 8. Both the sample exam and solutions to it are available on bCourses.

Solutions to the final exam itself have also now been posted on bCourses.

Homework Score

The overall homework score is now posted on bCourses.

Note that the homework score has not been computed on a percentage basis.

Instead, the aggregate homework score was computed as follows. First, the clicker scores were turned into two fake homework scores. (As it turned out, almost everyone got maximal clicker points, so this didn't really have an effect on the aggregate homework score.) Homework scores were first adjusted so that the mean on each homework (among those turned in) was the same across homeworks. Then the bottom three were dropped. The resulting scores were added. Scores in roughly the bottom half of the class were brought up, so that homework would not have a disproportionate effect on the final grade. The resulting score was adjusted to make its standard deviation closer to a number proportionate to the standard deviations of the midterms and finals (as if it were a hypothetical exam worth 150 points). I also made it so that the top score was 150.

As a result, the percentage you get by dividing your aggregate homework score by 150 will be approximately an increasing function of the average that you might compute on your own, but it will not be the same value, nor will it be linear.

Last updated 17 December 2021