Syllabus.

Math 110 – Number Theory, Winter 2015.
MWF 9:30 – 10:40 am, Engineering 2, 192
https://people.ucsc.edu/∼yorik/math110
Instructor: Yonatan Katznelson
Office: Baskin Engineering, 361 B
Phone: 459 - 1046
Email: [email protected]
Required text: Elementary Theory of Numbers by William J. LeVeque.
Course Description: Math 110 is an introduction to elementary number theory. Topics
include divisibility and the primes, congruence arithmetic, quadratic reciprocity, continued
fractions, Diophantine approximation and Diophantine equations. As time allows, we will
also explore some of the applications of number theory to cryptography.
Comment: The word ‘elementary’ as it applies to number theory, is not intended to mean
‘basic’ or ‘simple’.† Traditionally, elementary number theory refers to the study of number
theory absent the use of complex (or Fourier) analysis. Using analysis can make some proofs
easier (e.g., the prime number theorem).
Reading: Students are expected to read the book, early and often. By this I mean that
you should plan on reading the entire book‡ at least twice before the end of the quarter and
that you should stay ahead of the lectures (as described in the schedule that follows). There
will also be supplements to the textbook (posted on the course website) that you should also
read in detail two or three times.
Homework: Homework assignments will be posted on the course website throughout the
quarter, and collected in class on the dates specified. Late homework will not be accepted.
You should start working on the assignments early, as they will often complement the lectures. The lowest one or two homework grades will be dropped.§ Students are encouraged
to collaborate on the homework, but each student must write up his/her own work
for submission. Assignments (or portions thereof) that are illegible or otherwise incomprehensible will receive a grade of 0, or close to it.¶
Exams: There will be a midterm exam and a comprehensive final exam. The exam dates
are listed in the lecture schedule that follows.
Sections: Sections are not mandatory, but they are highly recommended. We cover a lot
of material at a fairly rapid pace, and mastering the material requires a great deal of study
and review. Sections provide the opportunity to review under the tutelage of an experienced
and knowledgeable TA.
†
Though some portions of the material in this course will be both.
With the exception of Chapter 6.
§
The number of dropped grades will depend on the total number of assignments collected.
¶
I.e., it is not enough to know how to do something—you have to express this knowledge clearly.
‡
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Course grade: Homework will contribute 40% to your score in the class, the midterm
contributes 20% and the final exam contributes the remaining 40%.
Letter grades will correspond (approximately) to the following ranges:
Overall Score
Grade
90 – 100
A− to A+
78 – 89
B− to B+;
60 – 77
C to C+
50 – 59
D
0 – 49
F
To pass the class, your overall score must be
60% or above.
Students with disabilities: If you qualify for classroom/exam accommodations because
of a disability, please submit your Accommodation Authorization Letter from the Disability
Resource Center (DRC) to me as soon as possible, preferably within the first week of the
quarter. Contact DRC by phone at 831-459-2089 or by email at [email protected] for more
information.
TIPS FOR SUCCESS
? Come to all the lectures and come prepared — read ahead in the book, so you have an
idea of what we will be discussing in the lecture. You are expected to master the methods
and concepts discussed in class, even if they are not in the book or the supplementary
notes.
? Go to section every week.
? Take advantage of all the resources: lecture, section and office hours.
? Study with friends for an hour or two a week. Technical skills can be practiced alone, but
concepts and proofs need to be discussed.
? The expected workload for a 5-unit course at UCSC is 15 hours a week, including lectures,
sections and studying outside of class — this means that you should be spending at least
8-10 hours a week studying outside of lecture/section.
To be successful, you should spread your studying over the week. Studying for five, six or
more hours in a row is not as effective as four or five blocks of 2 - 3 hours.
? If you feel that you are getting lost, take action. Come to office hours to clear up any
confusion or difficulty. Moreover, you should come to office hours prepared — the more
specific your questions the more helpful the answers will be.
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CHEATING:
Cheating in any form (using notes on exams, copying someone else’s homework, etc.) will not be tolerated. Any student caught cheating will be reported to
the Mathematics department and to his or her college
provost. In most cases, students caught cheating will receive a failing grade. Students who help others cheat are
also considered cheaters.
Cheating devalues everyone’s grades.
You should not tolerate it either.
Lecture Schedule with Exam Dates.
Topics marked with a ♣ are covered in supplementary notes on the course website.
Monday, 1-5: Introduction; What is number theory?
Wednesday, 1-7: Foundations and logic (methods of proof).
Friday, 1-9: Divisibility; division with remainder.
Monday, 11-12: The Euclidean algorithm and the GCD.
Wednesday, 1-14: The fundamental theorem of arithmetic (unique factorization).
Friday, 1-16: The distribution of primes, I.
Monday, 1-19:
Martin Luther King day.
Wednesday, 1-21: Linear Diophantine equations. Congruence.
Friday, 1-23: Arithmetic (mod m).
Monday, 1-26: Residue systems and reduced residue systems.
Wednesday, 1-28: Euler’s totient function, ϕ(n)
Friday, 1-30: Linear congruences and the Chinese remainder theorem.
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Monday, 2-2: Polynomial congruences; Lagrange’s theorem.
Midterm exam.
Wednesday, 2-4:
Friday, 2-6: Quadratic residues; Euler’s criterion.
Monday, 2-9: Quadratic reciprocity I.
♣
Wednesday, 2-11: Quadratic reciprocity II; Dirichlet’s theorem.
♣
Friday, 2-13: Powers and orders (mod m): Fermat’s theorem and Euler’s theorem.
Monday, 2-16:
Presidents’ day.
Wednesday, 2-18: Primitive roots and indices.
Friday, 2-20: Applications to cryptography: from Vig´en`ere to RSA.
Monday, 2-23: Cryptography II.
♣
♣
Wednesday, 2-25: Cryptography III.
♣
Friday, 2-27: Continued fractions I.
Monday, 3-2: Continued fractions II.
Wednesday, 3-4: Continued fractions III.
Friday, 3-6: Diophantine approximation I.
Monday, 3-9: Diophantine approximation II.
Wednesday, 3-11: Diophantine equations I: x2 + y 2 = z 2 .
Friday, 3-13: Diophantine equations II: x4 + y 4 = z 4 .
Monday, 3-15: Catch up and review.
Wednesday, 3-18:
Final Exam: 12:00 – 3:00 pm
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