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MASTER SYLLABUS

Master Syllabus

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Administrative Unit: Computer and Mathematical Sciences Department
Course Prefix and Number: MATH 305
Course Title: Number Theory
Number of:
Credit Hours 3
Lecture Hours 3
Lab Hours 0
Catalog Description: The goal of this course is to provide a modern treatment of number theory. The student learns more about integers and their properties, important number-theoretical ideas and their applications. The course emphasizes reading and writing proofs. Prerequisites: Grade of C or higher in both MATH 222 and MATH 225.
 
Prerequisite(s) / Corequisite(s): Grade of C or higher in both MATH 222 and MATH 225.
 
Course Rotation for Day Program: Offered even Fall.
 
Text(s): Most current editions of the following:

A Friendly Introduction to Number Theory
By Silverman, Joseph (Pearson)
Recommended
Elementary Number Theory
By Rosen, Kenneth R. (Addison-Wesley)
Recommended
Elementary Number Theory
By Burton, David M. (McGraw-Hill)
Recommended
Elementary Number Theory
By Eynden (Waveland Press)
Recommended
 
Course Objectives
  • To progress from a procedural/computational understanding of mathematics to a broad understanding encompassing logical reasoning, generalization, abstraction, and formal proof.
  • To convey mathematical knowledge in a variety of settings, both orally and in writing.
  •  
    Measurable Learning
    Outcomes:

  • Know the basic properties of integers such as divisibility, prime and congruence.
  • Apply the (extended) Euclidean algorithm (both iterative and recursive versions).
  • Apply the algorithms for modular arithmetic.
  • Solve linear Diophantine equations and linear congruences.
  • Solve a system of linear congruences by means of the Chinese Remainder Theorem.
  • Apply three famous results in congruences: Wilson’s theorem, Fermat’s little theorem and Euler’s theorem.
  • Determine the number of divisors of a positive integer (that is, the arithmetic function τ) and find their sum (that is, the arithmetic function σ).
  • Determine the number of integers between 1 and n that are relatively prime to n, that is the arithmetic function Φ.
  • State and apply properties of the three functions τ, σ and Φ.
  • State the meaning and significance of multiplicative functions.
  •  Apply results concerning primitive roots.
  • Apply the strong pseudoprime test.
  • Determine whether an integer is a quadratic residue.

  •  
    Topical Outline:

  • What is number theory?
  • Pythagorean triples
  • Pythagorean triples and the unit circle
  • Sums of higher powers and Fermat’s Last Theorem
  • Divisibility and the greatest common divisor
  • Linear equations and the greatest common divisor
  • Factorization and the Fundamental Theorem of Arithmetic
  • Congruences
  • Congruences, powers and Fermat’s Little Theorem
  • Congruences, powers and Euler’s Formula
  • Euler’s Phi Function and the Chinese Remainder Theorem
  • Prime numbers
  • Counting primes
  • Mersenne Primes
  • Mersenne Primes and perfect numbers
  • Powers modulo m and successive squaring
  • Computing kth roots and modulo m
  • Powers, roots and “unbreakable” codes
  • Primality testing and Carmicheal numbers
  • Euler’s Phi Function and sums of divisors
  • Powers modulo p and primitive roots 

  •  
    Culminating Experience Statement:

    Material from this course may be tested on the Major Field Test (MFT) administered during the Culminating Experience course for the degree. 
    During this course the ETS Proficiency Profile may be administered.  This 40-minute standardized test measures learning in general education courses.  The results of the tests are used by faculty to improve the general education curriculum at the College.

     

    Recommended maximum class size for this course: 25

     
    Library Resources:

    Online databases are available at http://www.ccis.edu/offices/library/index.asp. You may access them from off-campus using your CougarTrack login and password when prompted.

     
    Prepared by: Ann Schlemper Date: November 11, 2013
    NOTE: The intention of this master course syllabus is to provide an outline of the contents of this course, as specified by the faculty of Columbia College, regardless of who teaches the course, when it is taught, or where it is taught. Faculty members teaching this course for Columbia College are expected to facilitate learning pursuant to the course objectives and cover the subjects listed in the topical outline. However, instructors are also encouraged to cover additional topics of interest so long as those topics are relevant to the course's subject. The master syllabus is, therefore, prescriptive in nature but also allows for a diversity of individual approaches to course material.

    Office of Academic Affairs
    12/04