MATH 5330
                                                       Spring 2018

Instructor:    John Greene

Office:           168 SCC                   Phone:  726-6328

email:            jgreene@d.umn.edu

Hours:          10-10:50 MWF, 1-1:50 WThF, and by appointment

Text:             Fundamentals of Number Theory, William LeVeque.  This is a good
                      book, and it is cheap.  However, it can be a little hard to read at times.
                      I will also hand out lots of notes.  In fact, the course is mostly based
                      on these notes.

References:    A Friendly Introduction to Number Theory, 2nd edition, Silverman
                      Elementary Number Theory and its Applications, Rosen

Grades:         Grades will be based on a 200 point final, two 100 point tests and
                      100 points in the form of homework assignments.  A note on grading:
                      This is a 5000 level class.  I see it as aimed mostly at undergraduates
                      but appropriate for graduate students.  As such, the curve will be
                      based on undergraduates.
    
Tests:            The two tests tentatively set for the 5th and 11th weeks.  These will be outside
                      of class, probably on Thursday evenings.  The final exam is scheduled for Friday,
                      May 4, 2-3:55 PM.  Parts of the exams may be in take home form.

Computational  comment:
                      This course has a high computational component.  I will ask for frequent
                      computer searches for data on homework, from which you might be asked
                      to draw conclusions, with analysis also aided by the computer.  Much of this
                      work can probably be done using a spreadsheet, but most of it will probably
                      require Mathematica or Maple or Sage or some other similar program.  There
                      are various web applications that can do much or all of this work.  I will say
                      more about these kinds of problems as the need arises.

Topics:          Pythagorean triples
                      Unique Factorization and the Euclidean Algorithm
                      Modular Arithmetic
                      Primes and Perfect Numbers
                      Fermat, Euler, and Pseudoprimes
                      The RSA Crypto-System, discrete logarithm Crypto-Systems
                      Factorization Techniques, breaking Crypto-Systems
                      Quadratic Residues and Quadratic Reciprocity

                      Additional Material selected from the following:
                      Fermat’s Last Theorem,              Primality Testing
                      Calculating digits of π,                Other

Student outcomes and assessment:    By the end of this course, students should be able to:

                      1.  Understand the importance of unique factorization.
                      2.  Demonstrate mastery of the fundamental algorithms of number theory and
                           how to apply them.
                      3.  Analyze the performance of various factoring algorithms and the
                           implications for the security of various crypto-systems.
                      4.  Understand the more abstract ideas of number theory such as quadratic
                           residues and how they apply to other aspects of number theory.

    These outcomes will be assessed through homework (aspects 1, 2, 3, 4),
    Exam 1 (1, 2), Exam 2 (2, 3, 4) and the Final Exam (1, 2, 3, 4).

        
Notes:           Individuals who have any disability, either permanent or
                      temporary, which might affect their ability to perform in
                      this class should contact me as soon as possible so that I can
                      adapt methods, materials or tests as needed to provide for
                      equitable participation.

                      I was told to include this:  Academic dishonesty tarnishes UMD's
                      reputation and discredits the accomplishments of students. UMD is
                      committed to providing students every possible opportunity to grow
                      in mind and spirit. This pledge can only be redeemed in an
                      environment of trust, honesty, and fairness. As a result, academic
                      dishonesty is regarded as a serious offense by all members of the
                      academic community. In keeping with this ideal, this course will
                      adhere to UMD's Student Academic Integrity Policy, which can
                      be found at    http://www.d.umn.edu/conduct/integrity/student.html
                      This policy sanctions students engaging in academic dishonesty with
                      penalties up to and including expulsion from the university for repeat
                      offenders.