Math 5336 Number Theory Dr. Cordero
Class activities for October 7th, 2002
Topic: Primes and Their Distribution
∑ Page 44 # 1, 2, 3a, c, e, 4, 5a, 6a
∑ Page 50 # 1, 2, 5.
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.)
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.
∑ 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.
∑ 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