Ben Gurion University of the Negev
Department of Mathematics
NUMBER THEORY
Course number: 20116031; Spring Semester 2016
Instructor:
Prof. Ido Efrat
 Office: Mathematics Building, Room 106
 Office hours: Monday 1012
 Tel.: (08) 6461627
 Email: efrat@math.bgu.ac.il
Time and place:
Sunday 1113, Building 28, Room 301
Wednesday 911, Building 34, Room 2
Grading policy:
There will be a final exam. Its grade will be the grade of the course.
Exercises
Exercise 1 (Solution of problem 6)
Exercise 2
Exercise 3
Exercise 4 (Solution of problem 5)
Exercise 5 + encoded
telegram
Exercise 6
Exercise 7
Exercise 8
Course plan:
 Divisibility and prime numbers
 Greatest common divisors
 The fundamental theorem of arithmetic
 The existence of infinitely many primes and their density
 Primes in arithmetic progressions
 Formulas for prime numbers
 Perfect numbers
 Mersenne and Fermat primes
 Open problems in elementary number theory
 Congruences
 Arithmetic in Z/m
 The FermatEuler theorem
 The Chinese remainder theorem
 Wilson's theorem
 Representation of integers as sums of 2 squares
 The Hamiltonian quaternions and representation of integers as
sums of 4 squares
 The multiplicative group of Z/m
 The structure of the group
 Primitive roots and conditions for their existence
 periods in decimal expansions and the Artin conjecture
 Application: RSA public key encryption
 Quadratic residues
 The Legendre and Jacobi symbols
 Euler's formula
 Gauss' "Golden Theorem": the quadratic reciprocity law
 Primality tests
 Continued fractions
 Finite and infinite expansions
 Best approximation by rationals
 Applications: the Gregorian and Hebrew calendars,
piano tuning
 The FermatPell equation
 Archimedes' Cattle Problem
 Algebraic numbers and algebraic integers (as time allows)
 Basic arithmetic properties
 Liouville numbers
 Examples of transcendental numbers
 Hilbert's 7th problem
 The Schanuel conjecture
Recommanded Bibliography:
 D.M. Breton, Elementary Number Theory, Everyman's University,
Israel 2003 (In Hebrew)
 G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, Clarendon Press, Oxford 1960
 K. Ireland, M. Rosen,
A Classical Introduction to Modern Number Theory,
Springer, New York 1982

W. Narkiewicz,
Elementary and Analytic Theory of Algebraic Numbers,
Springer, Berlin 1990
 E. Grosswald,
Topics from the Theory of Numbers,
Birkhauser, Boston 1984

T. Apostol,
Introduction to Analytic Number Theory, Springer,
New York 1976
Some related links:
The prime pages
Fermat's last theorem
A Babylonian list of Pythagorean triples (approx. 1800 BC)
The Greatest Internet Mersenne Prime Search GIMPS
A biographies of
Euclid and Fermat
Euclid's "Elements" (D. Joyce)
To the Mathematics Department's home page