Prime Numbers

Date: 01/31/97 at 19:28:21
From: robby
Subject: prime numbers

What are the prime numbers and why are they prime numbers?

Date: 02/03/97 at 13:22:50
From: Doctor Reno
Subject: Re: prime numbers

Hi, Robby!

You've asked my favorite question! I love prime numbers, and spend 
a lot of time with my sixth grade class studying them.

Before we talk about prime numbers, though, Robby, I want to review 
some multiplication vocabulary with you...

Let's look at the multiplication problem 5 x 6 = 30. The 5 and 6 are 
called factors. They are the numbers we multiply together to get the 
answer 30, which is called the product. 30 is also called a multiple 
of 5 because it is a product of 5 and another number, 6. 30 is also a 
multiple of 6 because it is the product of 6 and 5. The number 5 has 
many other multiples, like 10, 15, 20, 25, 35, 40, etc. The number 6 
also has many other multiples, like 12, 18, 24, 36, 42, etc. Notice, 
Robby, that multiples are the "times tables" for a number. 30 is a 
multiple of not only 5 and 6, but also of 2 (2x15), 3 (3x10), 10, 
and 15.

A prime numbers is a number greater than one whose factors are only 
one and itself. In other words, 6 is not prime, because its factors 
are 1, 2, 3, and 6 (1x6, 2x3). But 5 is prime, because the only way 
you can get a product of 5 is by multiplying 1 and 5 (1x5). 

Composite numbers are all the other positive numbers greater than one. 
6 is composite.

The number 1 is not prime OR composite because it has only one factor. 

Mathematicians have been fascinated by prime numbers for thousands of 
years. In fact, Eratosthenes (275-194 BC, Greece), devised a "sieve" 
to discover prime numbers. A sieve is like a strainer that you drain 
spaghetti through when it is done cooking. The water drains out, 
leaving your spaghetti behind. Well, Eratosthenes's sieve drains out 
composite numbers and leaves prime numbers behind. To do what 
Eratosthenes did, make a chart of the first one hundred whole numbers 
(1-100):

The Sieve of Eratosthenes:

 1  2  3  4  5  6  7  8  9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100

Next, cross out 1, because it is not prime. Then, circle 2, because it 
is the smallest positive even prime. Now cross out every multiple of 
2; in other words, cross out every 2nd number.Then circle 3, the next 
prime. Then cross out all of the multiples of 3; in other words, every 
third number. Some, like 6, may have already been crossed out since 
they may be multiples of 2. Then circle the next open number, 5. Now 
cross out all of the multiples of 5, or every 5th number. Continue 
doing this until all the numbers through 100 are either circled or 
crossed out. Now, Robby, if you have remembered your multiplication 
tables, you have just circled all the prime numbers less than 100.

Computer people have written computer programs that do this same sieve 
of Eratosthenes. They use the sieve to test computers and to tell them 
how much faster one computer runs than another computer. You don't 
have to stop the sieve at 100 - you can go up as far as you want to 
find all the prime numbers you want to find. But, as you found, there 
is a lot of multiplication involved. 

Mathematicians are always trying to find large prime numbers. There 
are an infinite number of prime numbers, but they always want to know 
what the next largest one is that can be written down. They have found 
so many prime numbers now that only computers have the time and energy 
to look for the next highest prime number. In fact, the newest prime 
number is 420,921 digits long! If you want to see what it looks like, 
go to:

  a href="http://isthe.com/chongo/tech/math/prime/mersenne.html">http://isthe.com/chongo/tech/math/prime/mersenne.html

This prime number was discovered on November 13, 1996, by Joel 
Armengaud, a 29-year-old Parisian programmer, on a home computer much 
like yours. The number is too long to write out, so it is written in a 
special way, using exponents: 2^1398269 - 1.  This tells us to 
multiply two by itself (2x2x2x2...) 1,398,269 times! And after we do 
that, we are supposed to subtract 1 from that product.

There are many different kinds of prime numbers: twin primes, Germain 
primes, Fermat primes, and Mersenne primes. Mersenne primes are 
getting a lot of publicity now on the internet. 

You might also want to read all about prime numbers in the Dr. Math 
FAQ:

  http://mathforum.org/dr.math/faq/faq.prime.num.html   

Investigating prime numbers may also make you curious about perfect 
numbers, Fibonacci numbers, and many other numbers. I really hope 
that you enjoy investigating prime numbers - they really are fun!

-Doctor Reno,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/