MAT3201: Number Theory

In this course, integers are studied without use of techniques from other mathematical fields. Examples include: The Goldbach conjecture concerning the expression of even numbers as sums of two primes. The twin prime conjecture about the infinitude of prime pairs.  Fermat's last theorem (stated in 1637 but not proved until 1994) concerning the impossibility of finding nonzero integers x, y, z such that xn + yn = zn for some integer n greater than 2. This course includes the following topics: The Integers; GCD and Prime Factorization; Theory of Congruences; Multiplicative Functions; Applications of Number Theory; and Quadratic Residues. 

Course Objectives  
On successful completion of this course unit, the learners should be able to: 

• State and apply axioms about the integers 
• State and use the principle of finite induction 
• State and prove the division algorithm 
• Define a prime number and locate primes using the sieve of Eratosthenes 
• Use the prime number theorem 
• State, prove and apply the Euclidean algorithm 
• State and prove the fundamental theorem of arithmetic 
• Solve any linear Diophantine equation 
• Solve any linear congruence 
• State, prove and use the Chinese Remainder Theorem
• Perform divisibility tests of 2,3,5, 7, 9 and 11
• Check errors in strings
• State, prove and use theorems of Fermat and Wilson
• Work with multiplicative functions 
• Use the Caesar Cipher 
• Implement algorithms in this course on a microcomputer 

Expected Learning Outcome
This course unit is meant: 

• To help the student understand the techniques on divisibility, 
• To provide proper use of the Euclidean algorithm to compute greatest common divisors, 
• Factorization of integers into prime numbers, 
• To investigate the perfect numbers and congruences 
• To take the student through questions in number theory that can be stated in elementary number theoretic terms, but require very deep consideration and new approaches outside the realm of elementary number theory to solve. 
• To provide instruction that contributes to the learners’ abilities to think critically and solve real life problems, to reason mathematically and apply computational skills.