Math 5336 Number Theory Dr. Cordero

Class activities for October 7th, 2002

Topic: Primes and Their Distribution

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

· Page 44 # 1, 2, 3a, c, e, 4, 5a, 6a

· Page 50 # 1, 2, 5.

**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.)

- Side Trip into the E-zone…

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.

- Give examples of even numbers which are E-primes and even numbers which are not E-primes.
- Give examples of E-numbers which have at least two different factorizations into E-primes. Can you find an example which has more that two factorizations.
**Theorem**(Euclid): There is an infinite number of primes.**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.

**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