Ben Gurion University of the Negev

Department of Mathematics

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NUMBER THEORY

Course number: 201-16031; Spring Semester 2016

Instructor: Prof. Ido Efrat

• Office: Mathematics Building, Room 106
• Office hours: Monday 10-12
• Tel.: (08) 6461627
• E-mail: efrat@math.bgu.ac.il

Time and place:

Grading policy:

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:

1. 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
   
   
2. Congruences
   
   • Arithmetic in Z/m
   • The Fermat-Euler 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
   
   
3. 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
   
   
4. Quadratic residues
   
   • The Legendre and Jacobi symbols
   • Euler's formula
   • Gauss' "Golden Theorem": the quadratic reciprocity law
   • Primality tests
   
   
5. Continued fractions
   
   • Finite and infinite expansions
   • Best approximation by rationals
   • Applications: the Gregorian and Hebrew calendars, piano tuning
   • The Fermat-Pell equation
   • Archimedes' Cattle Problem
   
   
6. 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:

To the Mathematics Department's home page