Dr. Cordero
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