Number Theory
Chapter 3 – Part I
Prime Numbers
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!
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….
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 ….
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 √n.
Pf:
Let n = ab. Then 1 √n 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 …
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!
Number Theory
Def.: The function π(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:
a) π(10) b) π(25) d) Π(50)
Try to graph this function on [0, 100]!
Number Theory
Also, is there a way to express primes?
1) The only even prime is 2.
2) Every odd integer is of the form 2k+1.
3) 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.
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.
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 π(x) to x/log x approaches 1
as x grows without bound.
(See page 80 for some values ;)!)
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 !
Number Theory
Can you construct 6 consecutive
composite integers using Thm. 3.5?
Is this the smallest set of 6 consecutive
integers?
Number Theory
Some Conjectures about Primes
1) Bertand’s Conjecture:
For all positive integer n >1, there is a prime p
such that n 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!
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.
Number Theory
Reference:
Elementary Number Theory and its
applications
K. H. Rosen
Fifth Edition