Number Theory

1. Number Theory Chapter 3 � Part I Prime Numbers

2. Number Theory Def: A prime is an integer greater than 1 that is divisible by no positive integers other than 1 and itself. An integer greater than 1 that is not a prime is called a composite. There are many interesting questions about primes. Lot of them are open!

3. Number Theory Lemma 3.1 Every integer greater than 1 has a prime divisor. Pf: By Contradiction. Suppose there is a positive integer greater than 1 having no prime divisors. Then, since the set of positive integers greater than 1 with no prime divisors is nonempty. Then by the well-ordering property�.

4. Number Theory One Question is: How many primes? Theorem 3.1 There are infinitely many primes. Pf: By Contradiction. Suppose there are only finitely many primes �.

5. Number Theory Also: How to tell if a number is prime? Theorem 3.2 If n is a composite integer, then n has divisors which are less than vn. Pf: Let n = ab. Then 1 < a = b < n such that a = vn, for otherwise b = a > vn which implies that n = ab > n which is a contradiction. By Lemma 3.1, a has a prime divisor, which by Thm 1.8 must divide n �

6. Number Theory Theorem 3.2 implies The Sieve of Eratoshenes which is an algorithm to locate all primes less than a given positive integer. How does it work? Find all the primes that are less than 50!

7. Number Theory Def.: The function p(x), where x is a positive real number, denotes the number of primes not exceeding x. Use the Sieve above to answer the following questions: p(10) b) p(25) d) ?(50) Try to graph this function on [0, 100]!

8. Number Theory Also, is there a way to express primes? The only even prime is 2. Every odd integer is of the form 2k+1. Every odd integer can also be expressed as 4k+1 or 4k+3. The primes of the form 4k+1 are 5, 13, 17, 29,37 � and of the form 4k+3 are 3, 7, 11, � 4) Also, 3n +1, 7n + 4 � can also be considered.

9. Number Theory More Generally, we have Theorem 3.3 Dirichlet�s Theorem on Primes in Arithmetic Progressions. Suppose a and b are two positive integers not both divisible by the same prime. Then the arithmetic progression an + b with n = 1, 2, 3, � contains infinitely many primes.

10. Number Theory We also are interested in how the primes are occurring in the set of natural numbers. 2, 3, 5, 7, 11, 13, 17, 19, 23 � One observation is Theorem 3.4 The Prime NumberThm: The ratio of p(x) to x/log x approaches 1 as x grows without bound. (See page 80 for some values ;)!)

11. Number Theory Theorem 3.5 For any positive integer n, there are at least n consecutive composite positive integers. Pf: Consider the n consecutive pos. int. (n+1)! + 2, (n+1)! + 3, � (n+1)! + n+1 When 2 = j = n + 1, we know that j | (n+1)!, so by Thm. 1.9 we know j | (n+1)! + j. These are all composite ?!

12. Number Theory Can you construct 6 consecutive composite integers using Thm. 3.5? Is this the smallest set of 6 consecutive integers?

13. Number Theory Some Conjectures about Primes Bertand�s Conjecture: For all positive integer n >1, there is a prime p such that n < p < 2n. (proven by Chebyshev 1852) 2) Twin Prime Conjecture: There are infinitely many pairs of primes p and P+2. (or twin primes such as 3 and 5, 5 and 7 �) (open problem)

14. Number Theory 3) Goldbach�s Conjecture: Every even positive integer greater than 2 can be written as the sum of two primes. 4) The n2 + 1 Conjecture: There are infinitely many primes of the form n2 + 1, where n is a positive integer. Let�s do some problems Page 86/3, 5, 6a, 10a, 12,

15. Number Theory Greatest Common Divisors � gcd Def: The gcd of two integers a and b, a,b >0, denoted by (a,b) or gcd(a,b), is the largest integer that divides both a and b. Example: Find gcd(48, 96). Def: If gcd(a,b) = 1, then we say a and b are relatively prime. Give an example!

16. Number Theory Theorem 3.6 Let a and b be integers with (a,b) = d. Then (a/d, b/d) = 1. Pf: Easy! Assume (a/d, b/d) = e and show that e = 1.

17. Number Theory Reference: Elementary Number Theory and its applications K. H. Rosen Fifth Edition