-function
Course meets Tu, Th 2:20-3:40 in Math 5-127.
I. Kra office hours (Th, January 25- W, May 9) in Math 4-111: Tu 11-12, W
3-4; in Undergraduate Mathematics office: Th 11-12; and by
appointment.
Text book
K.H. Rosen
ELEMENTARY NUMBER THEORY and its applications
Fourth Edition
There are many excellent undergraduate books on the subject. I
have placed the following ``sample'' on reserve in the
Mathematics/Physics/Astronomy Library:
The books are a mixture of classical texts (for example,
Dirichlet) and modern efforts (for example, LeVeque), elementary (for
example, Silverman) and advanced (for example, Ireland and Rosen), algebraic
(for example, Andrews) and analytic approaches (for example, Apostal). The
course will concentrate on elementary algebraic number theory. More advanced
analytic topics (for example, the Prime Number Theorem) require a function
theoretic (complex analysis) background.
Examinations There will be a half hour pretest (on Th 2/1)
that will cover very basic material from the algebra courses that are
prerequisites for this course,
two midterm examinations (on Tu 2/20 & Th 4/5) and a final examination (on
Th 5/10). There will also be 3 un-announced twenty minute quizzes.
Projects and homework Homework is an integral part of the
course. Of the 15 problem sets (of even numbered exercises), 12 are to be
handed in (on first meeting (usually on a Tuesday) following the assignment);
the homework grade will be based on the best 10 of these. In
addition students will hand in 2 of the 15
research/scholarship/computing
projects. The projects with a nontrivial writing component can be used to
satisfy the Mathematics Upper Division Writing Requirement.
Grading The final examination will constitute 25% of the
grade; each of the midterms, 15%; the pretest 10%; the quizzes, 5% each;
each of the two projects, 5%; the homework 10%. The tests and quizzes
will be constructed so that at least 50% of each will consist of statements of
definitions, description of key results and routine calculations. A total
grade of at least 60% will be required for a C; 90% will guarantee an
A.
Special needs If you have a physical, psychiatric, medical or
learning disability that may impact on your ability to carry out assigned
course work, you may contact the Disabled Student Services (DSS) office
(Humanities 133, 632-6748/TDD). DSS will review your concerns and determine,
with you, what accommodations may be necessary and appropriate. I will take
their findings into account in deciding what alterations in course work you
require. All information on
and documentation of a disability condition should be supplied to me in
writing at the earliest possible time AND is strictly confidential. Please
act early, since I will not be able to make any retroactive course
changes.
| Topic | Practice problems | Hand in | |
| Th 1/25 | 1.1 Numbers, sequences, and sums | p14/1,5,9,25,33 | 2,4,26,28 |
| Wk of 1/30 | 1.2 Mathematical induction | p22/3,13,15 | 10,16,30 |
| 1.3 The Fibonacci numbers | p28/3,7,25,29 | 4,8,22 | |
| 1.4 Divisibility | p34/3,7,11,29,58 | 4,6,16,30 | |
| 3.1 Prime numbers | p76/3,5,9,13,29 | 6,10,16 | |
| Th 2/1 | Half hour pretest | ||
| Wk of 2/6 | 3.2 Greatest common divisors | p84/13,15,31 | 20,22,32 |
| 3.3 The Euclidean algorithm | p94/5,7,19 | 20,22 | |
| 3.4 The fundamental theorem of arithmetic | p104/3,11,19 | 6,12,20 | |
| Wk of 2/13 | 3.6 Linear Diophantine equations | p123/3,9,21 | 10,20,22 |
| 4.1 Introduction to congruences | p135/5,17,25 | 8,20 | |
| 4.2 Linear congruences | p141/1,7,15 | 4,10,16 | |
| Tu 2/20 | First midterm (up to §4.2) | ||
| Th 2/22 | 4.3 The Chinese remainder theorem | p149/5,15,21 | 6,14,24 |
| 4.4 Solving polynomial congruences | p158/1,3,11 | 10,12 | |
| Wk of 2/27 | 4.5 Systems of linear congruences | p167/1,3,5,7 | 2,4,12 |
| 5.1 Divisibility tests | p177/3,11,19 | 4,12,26 | |
| Th 3/1 | First project due | ||
| Th 3/8 | 6.1 Wilson's and Fermat's little theorem | p202/1,3,17 | 14,18,34 |
| 6.2 Pseudoprimes | p213/3,9,11 | 6,20 | |
| Wk of 3/13 | 6.3 Euler's theorem | p218/1,5,19 | 2,6,20 |
| 7.1 The Euler -function |
p237/1,3,19,21 | 4,8,22,26 | |
| 7.2 The sum and number of divisors | p235/3,5,21 | 4,6,8 | |
| Wk of 3/27 | 7.4 Möbius inversion | p256/3,9,17,23 | 6,10,24 |
| 8.1 Character ciphers | p266/3,15 | 14,16 | |
| 8.2 Block & stream ciphers | p278/13,19 | 14,18 | |
| 8.3 Exponentiation ciphers | p284/3,5 | 4,6 | |
| Tu 4/3 | 8.4 Public key cryptography | p290/1,3,7 | 2,6,12 |
| 9.1 The order of an integer and primitive roots | p313/3,11,21 | 6,10,20 | |
| Th 4/5 | Second midterm (up to §8.2) | ||
| Wk of 4/10 | 9.2 Primitive roots for primes | p318/7,13,15 | 8,12,16 |
| 9.3 The existence of primitive roots | p328/1,5,9 | 2,12,14 | |
| Wk of 4/17 | 9.4 Index arithmetic | p337/3,9,21 | 4,10,22 |
| Wk of 4/24 | 11.1 Quadratic residues and nonresidues | p386/5,9,15,41 | 6,8,18 |
| Tu 5/1 | Second project due | ||
| Wk of 5/1 | 11.2 The law of quadratic reciprocity | p401/3,7,11 | 2,8,12 |
| Tu 5/8 | 11.3 The Jacobi symbol | p410/3,7,11 | 4,8,10 |
| 11.4 Euler pseudoprimes | p419/1,3,7,9 | 2,4,6 | |
| Th 5/10 | Final examination (2:00 to 4:30) |
Choose one of the following projects to complete hand in by May 1.
A proof that 
or 
is transcendental.
Programming project 6.1.2.
C P E 7.1.4
Programming project 7.4.1.
Programming project 9.3.2.
Programming project 11.1.3.
A short (2 to 4 typed pages) paper on the mathematical contributions of a
number theorist.
NOTE: For a programming project, hand in an outline of the program, the code
for the program, and a reasonable amount of program output.