An integer (greater than one) is prime if the only whole numbers it can be divided by (without a remainder) are itself and one. All other integers are composite. In other words, a prime number has only two positive factors. Composite numbers have more. For example, seven is a prime number because its only positive factors are one and seven. Fifteen is composite because it has four: one, three, five, and fifteen.
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Eratosthenes' Prime Number Sieve5 stars
Eratosthenses was a Greek mathematician who figured out that to find all the prime numbers between two and some large number, you need to remove all the multiples of each number between two and your large number. Start by pressing "2" (skip over "1"), and you'll see all the multiples of two eliminated: 2,4,6,8, etc. Next, click on "3" and so on. At some point the program will stop, and all the prime numbers between 2 and 400 will be colored red. Can you guess the biggest number you will need to click?

Dr. Math: Prime Numbers4 stars
"A prime number is a positive integer that has exactly two positive integer factors, 1 and itself. For example, if we list the factors of 28, we have 1, 2, 4, 7, 14, and 28. That's six factors. If we list the factors of 29, we only have 1 and 29. That's 2. So we say that 29 is a prime number, but 28 isn't." Dr. Math presents an excellent introduction to prime numbers, the Sieve of Eratosthenes, and links to other prime number sites.

FactMonster: Prime Numbers4 stars
Fact Monster begins with a short prime number lesson, and a table of all the prime numbers between 1 and 1000. On the next page ("World's Largest Known Prime Number") is a simple explanation of Mersene primes, and the search for bigger and bigger primes. Although there are an infinite number of primes, it is only with today's computing power can we actually name them. In fact, the Electronic Frontier Foundation is offering is a $100,000 reward for finding a prime number with at least 10 million digits.

Math Forum: Prime Numbers4 stars
For middle school and high school students, this Math Forum goes behind simple prime number definition, and introduces both Euclid's theory of prime numbers (which has been proven) and Goldbach's Conjecture (which hasn't.) "In a letter to Leonard Euler in 1742, Christian Goldbach conjectured that every positive even integer greater than 2 can be written as the sum of two primes. Though computers have verified this up to a million, no proof has been given.

The Prime Pages5 stars
Our last site of the day is the most comprehensive. It is not for elementary students just being introduced to prime numbers, but rather for serious high school and college math students who want to explore current projects in number theory. Best clicks are the Prime Glossary, Brief History of Large Prime Numbers, and Prime Curios ("an exciting collection of curiosities, wonders and trivia related to prime numbers.")
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Feldman, Barbara. "Prime Numbers." Surfnetkids. Feldman Publishing. 17 Nov. 2004. Web. 24 Apr. 2014. <http://www.surfnetkids.com/resources/primenumbers/ >.
By Barbara J. Feldman. Originally published November 17, 2004. Last modified November 17, 2004.
Honorable Mentions
The following links are either new discoveries or sites that didn't make it into my newspaper column because of space constraints. Enjoy!
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