MTH 3214: Number Theory, 3CU
Pre-requisites: None
Course Description
In this course, integers are studied with little use of techniques from other mathematical fields.
Questions of divisibility, use of the Euclidean algorithm to compute greatest common divisors,
factorization of integers into prime numbers, investigation of perfect numbers and congruences
belong here. Some important discoveries of this field are Fermat's little theorem, Euler's
theorem, and the Chinese remainder theorem. The properties of multiplicative functions such as
the Möbius function, and Euler's φ function also fall into this area. The course will 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. 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.
Course Objectives
By the end of this course, the student should be able to:
 State axioms about the integers
 State and apply the principle of finite induction
 State and prove the division algorithm
 Define a prime number and locate primes using the sieve of Eratosthenes
 State and prove the Euclidean algorithm
 State and prove the fundamental theorem of arithmetic
 Solve a linear Diophantine equation
 Solve a linear congruence
 State and prove the Chinese Remainder Theorem
 Perform divisibility tests of 2,3,5, 7, 9 and 11
 Check errors in strings
 State and prove theorems of Fermat and Wilson
 Use the law of quadratic reciprocity.
Detailed Course Outline
The Integers
Basic properties, summation and products, mathematical induction, divisibility, Primes, Sieve of
Eratosthenes, Prime number theorem, Prime producing functions, some conjectures on primes.
(10hrs)
GCD and Prime Factorization
Greatest Common Divisor, Euclidean Algorithm, Fundamental Theorem of Arithmetic, LCM,
direct factorization method, Fermat factorization, Fermat numbers and primes, Linear
Diophantine equation and its solution. (10hrs)
Theory of Congruences
Introduction to congruences, properties, residues modulo m, Algebra of Congruences, Linear
congruences, special congruences, The Chinese Remainder theorem, Systems of Linear
congruences, Divisibility tests, check digits, Theorems of Wilson, Euler and Fermat. (9hrs)
Multiplicative Functions
The Euler Phi-function, the sum and number of divisors, Perfect numbers and Mersenne primes.
(6hrs)
Applications of Number Theory
Character or Monographic ciphers. (4hrs)
Quadratic Residues
Legendre symbol, Quadratic Reciprocity, Jacobi’s symbol, The order of an integer, Primitive
roots. (6hrs)
Reading List
The reading list will include but is not limited to the following texts.


Burton, D.M (1976). Elementary Number Theory. Allyn and Bacon.
Kasozi, J. and Mangheni, P.J. Number Theory. Department of Distance Education,
IACE, Makerere University. (in print)
Hardy, G.H and Wright, E.M (1979). An Introduction to the Theory of Numbers. 5th Edition,
Oxford University Press.
Learning Outcome
State axioms about the integers
State and apply the principle of finite induction
State and prove the division algorithm
Define a prime number and locate primes using the sieve of Eratosthenes
State and prove the Euclidean algorithm
State and prove the fundamental theorem of arithmetic
Solve a linear Diophantine equation
Solve a linear congruence
State and prove the Chinese Remainder Theorem
Perform divisibility tests of 2,3,5, 7, 9 and 11
Check errors in strings
State and prove theorems of Fermat and Wilson
Use the law of quadratic reciprocity.