Math 5336 Number Theory                                                       Dr. Cordero

 

Class activities for October 7th, 2002

Topic: Primes and Their Distribution

 

  1. Discuss homework from last time.
  2. What is a prime number? A composite number?
  3. Use the Sieve of Eratosthenes to obtain all the prime numbers less than 100.
  4. Determine whether 7679 is prime. If it is, say why it is. If itís not, give the complete factorization into primes.
  5.  Practice:

        Page 44 # 1, 2, 3a, c, e, 4, 5a, 6a

        Page 50 # 1, 2, 5.

  1. Fundamental Theorem of Arithmetic

Every positive integer n>1 can be expressed as a product of primes; this representation is unique, apart from the order in which the factors occur.

(Proof given on page 42.)

  1. Side Trip into the E-zoneÖ

Let E be the set of positive even integers. An even integer is said to be an E-prime if it is not the product of two other even integers. Equivalently, an even integer is an E-prime if it has no E-factors other than itself.

  1. Theorem (Euclid): There is an infinite number of primes.
  2. Twin Prime Conjecture

        There are several examples of pairs (p, p+2) where both p and p+2 are prime. Give examples.

        Such pairs are called twin primes.

        The Twin Prime Conjecture: There are infinitely many pairs of twin primes.

  1. The Goldbach Conjecture

        Can every even number (greater than 2)  be written as a sum of two primes? Give examples.

        The Goldbach Conjecture: Every even number greater than 2 can be written as a sum of two primes.

         Practice: page 59 # 1, 2, 3, 19