Announcements: I do not use CANVAS. This page will be the only online source for this course. All announcements will be posted here.
Things to know for the final exam: Definitions/notions: basis representation of integers, divisibility, greatest common divisor, least common multiple, co-primality of integers, integer ideal, principal ideal, prime numbers, rearrangements and combinations, binomial coefficients, generating function, congruence, complete/reduced residue systems, linear and polynomial congruences, arithmetic functions (Euler phi function, d(n), sigma(n), Mobius function), multiplicative functions, Mobius pair, exponent of a residue class, primitive root, quadratic residue, Legendre/Jacobi symbols; Statements of Theorems: Basis representation theorem, Euclid’s division lemma, Fundamental Theorem of Arithmetic, Solution set of Diophantine equation, Binomial Theorem, Fermat’ Little Theorem, Wilson’s Theorem, characterization of complete/reduced residue systems, Euler’s theorem, Chinese Remainders Theorem, formulas relating various arithmetic functions, Mobius inversion formula, Euler’s Criterion, Quadratic Reciprocity Law.
Prerequisites: MAC 2312, MAC 2512 or MAC 3473 with a minimum grade of C; MAS 3300 or MHF 3202 recommended.
Time and Room: MTWRF 5 (2:00-3:15 p.m.), LIT 127.
Textbook: Number Theory by George E. Andrews, Dover Publications, New York, 1994. ISBN-13: 978-0-486-68252-5, ISBN-10: 0-486-68252-8.
Critical dates: Jul. 1 (classes begin), Aug. 9 (classes end). Exams: M1 – Jul. 12, M2 – Jul. 26, F – Aug. 9. Holidays: Jul. 4 (Independence Day).
Office Hours: MTRF 4 (12:30-1:45 p.m.) in LIT 458, or by appointment made 24 hours in advance. Please, call me at 352-294-2326 or use e-mail: pilyugin@ufl.edu for communication. For more details, see my summer schedule.
Description and Objectives of the Course: This course introduces basic concepts of analytical number theory. The course content is as follows:
W1: Introduction, notes on mathematical induction, basis representation theorem,
divisibility, linear Diophantine equations, Fundamental Theorem of Arithmetics, elementary concepts of combinatorics.
W2: Fermat’s little theorem, Wilson’s theorem, generating functions, congruences and their properties, residue systems, linear congruences, the Chinese Remainder theorem, polynomial congruences, midterm 1.
W3: Arithmetic functions (phi,d,sigma) and their properties, multiplicative arithmetic functions, Mobius inversion formula, properties of reduced residue systems.
W4: Prime numbers, Tchebyshev’s theorem, introduction to quadratic congruences, midterm 2
W5: Elementary partition theory, Euler’s partition theorem, partition generating functions, identities between infinite sums and products, examples of partition identities.
W6: Algebraic concepts: rings and fields, finite fields, residue systems revisited, Gaussian integers, Hamiltonian ring of quaternions, the Four Squares theorem, final exam.
Grading System: The grade is determined by the cumulative score on 2 midterms (30 points each) and the final exam (40 points). The letter grade is determined according to the following table
Letter Grade
A
A-
B+
B
B-
C+
C
C−
D+
D
Score
100 – 93
92 – 88
87 – 83
82 – 74
73 – 69
68 – 64
63 – 59
58 – 54
53 – 49
48 – 40
Course policies:
Closed-book policy: No use of calculators, or books will be allowed during in-class tests.
Grading disputes: Any issues or questions about the grading of exams must be brought to the instructor’s attention within one week after the exams are returned to the class.
Excused absences: In certain circumstances a student will be able to make up a missed exam. These circumstances could include medical situations, family emergencies, travel for University activities (eg. band, debating club, etc), and religious observances. In these cases the student must inform the instructor before or within one week after the missed work and provide written documentation.
Policy on class attendance: Requirements for class attendance and make-up exams, assignments, and other work in this course are consistent with university policies that can be found at: https://catalog.ufl.edu/ugrad/current/regulations/info/attendance.aspx
Most students benefit a great deal from attending class regularly. Arriving late and/or leaving early, reading the newspaper, looking at your cell phone, etc. disrupts the class and is rude and unprofessional.
UF Honor Code: “UF students are bound by The Honor Pledge which states: “We, the members of the University of Florida community, pledge to hold ourselves and our peers to the highest standards of honor and integrity by abiding by the Honor Code. On all work submitted for credit by students at the University of Florida, the following pledge is either required or implied: “On my honor, I have neither given nor received unauthorized aid in doing this assignment.” The Honor Code specifies a number of behaviors that are in violation of this code and the possible sanctions. Furthermore, you are obligated to report any condition that facilitates academic misconduct to appropriate personnel. If you have any questions or concerns, please consult with the instructor or TA’s in this class.”
For students with disabilities: “Students with disabilities requesting accommodations should first register with the Disability Resource Center (352-392-8565, https://www.dso.ufl.edu/drc/) by providing appropriate documentation. Once registered, students will receive an accommodation letter which must be presented to the instructor when requesting accommodation. Students with disabilities should follow this procedure as early as possible in the semester.”
Online evaluations: Students are expected to provide feedback on the quality of instruction in this course based on 10 criteria. These evaluations are conducted online at https://evaluations.ufl.edu. Evaluations are typically open during the last two or three weeks of the semester, but students will be given specific times when they are open. Summary results of these assessments are available to students at https://evaluations.ufl.edu/results.
Contact information for the Counseling and Wellness Center: https://counseling.ufl.edu/, 392-1575; and the University Police Department: 392-1111 or 9-1-1 for emergencies.