Mathematics 350 (3 credit hours)
Number Theory (Spring 2011)
www.utm.edu/~caldwell/classes/251/

This page: [ Prerequisites | Catalog Description | Grade | Presentations | Objectives | Text | Outline  ]
Related pages: [ The Prime Pages | Number Theory Web | RSA FAQ | MathSciNet | Integer Sequences | Listen to Primes ]

Teacher:
Dr. Caldwell, office 429 Holt Humanities, phone 7336. Department office 7360. Email: caldwell@utm.edu.
Prerequisites:
Mathematics 314 (aka 241) 
Catalog Description:
The integers: well-ordering, different bases, divisibility, primes, and factoring.  The fundamental theorem of arithmetic and the division algorithm.  Diophantine equations and applications of congruences.  Pseudorandom numbers, pseudoprimes, and cryptography. 
Grade:
Grading will be done according to the following weights in a "fair and subjective" manner 
  • 40 %   tests (3 one-hour tests) 
  • 25 %   homework (assigned daily) 
  • 20 %   final (comprehensive) 
  • 15 %   classroom participation (see below)
Homework:
Homework is due at the beginning of the following class meeting (it must be on the table before class begins).  Homework may be turned in early (place it in my mail box, my hand, or gently slide it under my door).  Late homework will be reduced in value by 50% for each day, or fraction thereof, it is late.  Some of the homework will be easy, some difficult and some may be impossible (because that is the way problems are in real life)!  Part of what you are to learn is what you can and cannot do.
Comments:
Number theory is the study of the integers which includes such things as cryptology, divisibility rules, finding massive primes...  It is a great course for secondary education students because of its many simple and curious problems.  Yes, number theory has proofs, though it is nowhere as proof intensive as Abstract Algebra or Real Analysis.
Departmental
Objectives:
The student will: 
  • Identify and apply various properties of and relating to the integers including the Well-Ordering Principle, primes, unique factorization, the division algorithm, and greatest common divisors.
  • Identify certain number theoretic functions and their properties.
  • Understand the concept of a congruence and use various results related to congruences including the Chinese Remainder Theorem.
  • Solve certain types of Diophantine equations.
  • Identify how number theory is related to and used in cryptography. 
Text:

Elementary Number Theory (5th Edition, 2004) by Kenneth H. Rosen, Addison Wesley, 744 pages, ISBN-10: 0321237072, ISBN-13: 978-0321237071.

Outline:
To be determined based on student/teacher interests, one possibility follows.  
  topic days
  The Integers 5
  Primes Greatest Common Divisors 6
  Congruences 5
  Some Special Congruences 5
  Multiplicative Functions 5
  Cryptology 6
  Other (e.g., primality proofs, "mathemagic") 5
  37
  One Hour Tests 3
  40