8.4: Prime Numbers
look for all the prime numbers from 2 to 30 next Carry out the following process of circling/crossing out numbers
circle the next number N that has not been crossed out and then cross out every Nth number after it until every number in the list has either been circled or crossed out. the circled numbers in the list are the Pos pg 331
it may happen that the number you are checking is not prime consider 899
when you divide 899 by consecutive PNs you will find that none of these primes divides 899 until you get to 29. Then you will discover that 899 = 29 x 21, so 899 is not prime
Euclid
300 B.C, another mathematician who lived in Ancient Greece wrote the element which proves infinitely many prime numbers
how can we find prime numbers what is sieve of Eratosthenes
Eratosthenes a mathematician and astronomer in Ancient Greece, discovered a method of finding/listing PN
how do we write numbers as products of prime numbers
a convenient way to factor a counting number into a product of prime numbers is to create a factor tree
use Euclid's reasoning to justify why prime numbers go on forever
a list of n prime numbers, consider the number p1 x p2 x p3 .........pn + 1, which is just the product of the list of PN with the number 1 added on this new number is not divisible by any of the primes on our list because each of these leaves a remainder of 1 when our new number is divided by any of these numbers either PN is a prime number that is not on our list of primes or when we factor our number into a product of prime numbers, each of these prime factors is a new prime number that is not on our list no matter how many prime numbers we started with, we can always find new PN, and it follows that there must be infinitely many prime numbers
it is not possible for
a product of prime numbers to be equal to a product of some different prime numbers (11, 13, any primes other than 5, 7, 23, 29) there is only one way to factor a cn that is not prime into a product of prime numbers
prime numbers or simply the primes
are the counting numbers other than 1 that are divisible only 1 and themselves counting numbers that cannot be factored in a nontrivial way 2 3, 5, 7
the prime numbers are to the counting numbers as
atoms are to matter
prime numbers
building blocks of the counting numbers
composite numbers
counting numbers other than 1 that are not prime ex: 6 is composite because 6=2 x 3
what is a factor tree
diagram which shows how to factor a number, how to factor the factors, how to factor the factors or factors, etc, until prime numbers are reached numbers at bottom of branches of factor tree are the prime factors of original number rearrange factorization by placing identical primes next to each other use notation of exponents
what if we don't see a way to factor a number
divide it by consecutive Prime numbers until we find a PN that divides it evenly.
to determine whether a number is prime by the trial division method,
divide the number by consecutive prime numbers starting at 2. if none of the PN divide your number, if you reach a point when the quotient becomes smaller than the prime number by which you are dividing, the the previous reasoning tells us the number must be prime
fundamental theorem of arithmetic
every counting number greater than 1 can be factored as a product of prime numbers
the prime numbers are considered to be the building blocks of the counting numbers. why? 145 = 29x5 2009 = 41x7x7 264, 264= 13 x 11x11x7x3x2x2x2
every counting number greater than or equal to 2 is either a prime number or can be factored as a product of prime numbers all counting numbers 2 onward are built from prime numbers
why are the circled numbers produced by the sieve of Eratosthenes the Pos in the list
if a number is circled, then we did not cross it out, using one of the previously circled numbers the numbers we cross out are exactly the multiples of a circled number (beyond that circled number) when we circle 3 and cross out every 3rd number after 3, we cross out the multiple of 3 beyond 3 (6, 9, 12, 15, etc) the circled numbers are the numbers that are not multiples of any smaller number other than 1. it is divisible only by 1 and itself and so is a prime number
why don't we have to check other numbers, such as 4 6 8 9 which are not prime numbers to see if they divide the given number?
if a number is divisible by some CN other than 1 and itself, then according to the fundamental theorem of arithmetic, that divisor is either a prime number or can be factored into a product of prime numbers, and each of those prime factors must also divide the number in question so to determine whether a number is prime, find out only whether any prime numbers divide it
why is 1 not included as a prime number
if we did, we'd have to restate the fundamental theorem of arithmetic in a complicated way
how do you use the method
list all the whole numbers from 2 up to wherever you want to stop looking for PN
mathematicians through the ages have been fascinated by PN. does the list of PN go on forever does it ever come to a stop
list of PN does go on forever, not obvious why
when we use trial division to determine whether a number is prime, how do we know when to stop dividing by primes? see if 283 is prime or not (divide by consecutive PN; we find that none of these PN divide 283) how do we know when we can stop dividing
look at the quotients that result when we divide 283 by PN in order as we divide by larger Pos, the quotients get smaller. the quotient becomes smaller than the divisor. if we continued dividing by larger numbers, the quotients will become smaller than division if a prime number larger than 17 were to divide 283, the corresponding quotient would be a whole number less than 17 and thus would have a prime number divisor less than 17. but we already checked all the PN up to 17 and found that none of them divide 283; we can tell that 283 is prime
trial division
method for determining whether a number is prime
prime numbers might seem to be only
of theoretical interest, have important practical applications to encryption ex: secure web Site on the internet, prime numbers are involved
factor 26, 741
pg 333
make a factor tree for 600
pg 333
how can you tell whether a number such as 239 is a prime number?
use the sieve method to find all prime numbers up to 239 (too slow) trial division: divide number by consecutive prime numbers (faster way) if you ever find a prime number that divides your number, then your number is not prime; otherwise it is prime
if two people made two different factor trees,
we still end up with same prime factors (fundamental theorem of arithmetic)
