Lecture 2: The Euclidean Algorithm

Background Material

• Euclidean division (division with remainder). a = q*b + r, where 0 <= r < b. The quotient q and remainder r are unique.
• Definition. g = gcd(a,b) iff g|a and g|b (m|n iff n = q*m for some q) and if e|a and e|b then e|g.
• Factorization into primes (every integer can be written uniquely (upto order) as a product of prime numbers. This is called the fundamental theorem of arithmetic. See lecture 1 on Unique Factorization.

Reading

• Chapter sections 1.3-1.4 of the text.

Topics

• Definition of greatest common divisors
• Euclidean algorithm
  gcd(a,b)
  1. if b = 0 then return a;
  2. else return gcd(b,a mod b)
• Proof of correctness
• The extended Euclidean algorithm (recursive version)
• Iterative version of the algorithm and remainder sequences
• Fibonacci numbers and an upper bound on the number of divisions
  • The Fibonacci numbers are defined by F_0 = 0, F_1 = 1, F_n = F_{n-1} + F_{n-2} for n > 1.
  • F_n = 1/sqrt(5)*(phi^n - phihat^n), where phi = (1 + sqrt(5))/2 and phihat = (1 - sqrt(5))/2 are the roots of x^2-x-1=0. F_n is the nearest integer to 1/sqrt(5)*phi^n.
  • Let 0 <= b < a <= N and n = the maximum number of division steps required to compute gcd(a,b) using the Euclidean algorithm. Then n <= 2lg(N). This constant in this bound is not tight.
  • Let 0 <= b < a <= N. The maximum number of division steps occurs when when a = F_{n+2} and b = F_{n+1}. From this it follows that the n <= log[phi](N) + log[phi](sqrt(5)/phi). log[phi](N) is approximately 1.44*lg(N).
• The probability that two random integers are relatively prime.

Maple worksheets and other resources

• euclid.mw - Maple worksheet on the Euclidean algorithm.
• euclid.pdf - Lecture notes on the Euclidean algorithm (to be posted).