THE SIEVE OF ERATOSTHENES: Prime and Composite Numbers by Jan Harrison

Eratosthenes Eratosthenes was a Greek scholar who lived in the 3rd century B.C. During his life, he made many important contributions to the fields of mathematics, geography, philosophy, and astronomy.

Among his contributions to mathematics (over 2,000 years ago), Eratosthenes is particularly well known for: Calculating a surprisingly accurate measurement of the Earth’s circumference. Devising the Sieve of Eratosthenes, a tool for finding prime numbers.

Prime Numbers A number is regarded as prime only if it has exactly two factors: 1 and itself. For example: The number 2 has exactly two factors: 1 and itself (i.e., 1 and 2); therefore, it is a prime number.

Composite Numbers If a number is not prime, then it is composite. For example: The number 12 has six factors: 1, 2, 3, 4, 6, and itself (i.e., 1, 2, 3, 4, 6, 12); therefore, it is a composite number.

The number 1: is neither prime nor composite. Why do you think this is? [Hint: Look back at the defini- tion of a prime num- ber.]

Prime or Composite? Is the number 22 prime or composite? Why? What about the number 39? 7?77?

The answers: 22 is a composite number, as it has more than two factors (1, 2, 11). 39 is a composite number, as it has more than two factors (1, 3, 13). 7 is a prime number, as it has only two factors (1, 7). 77 is a composite number, as it has more than two factors (1, 7, 11).

2 is the only even prime number. Why?

Building Blocks Have you noticed that the factors of composite numbers are always prime numbers? The Fundamental Theorem of Arithmetic states that every natural number greater than 1 has one unique way of being represented as a product of its prime factors. For this reason, the prime numbers are sometimes called the “building blocks” of the natural numbers.

Finding Prime Numbers The Sieve of Eratosthenes is a tool that will help you find prime numbers. [Note: A worksheet is available at the web page for this lesson, if you want to try this as you read along.] worksheet

Image borrowed from mber2003/Sieve.html The Sieve of Eratosthenes generally looks something like this (depending on the range of numbers included):

How to use the Sieve of Eratosthenes: 1) 1) Since the number 1 is neither prime nor composite, draw a box around 1. 2) 2) Circle the first unmarked number, 2, which is the first prime number. Then count by 2, crossing out each multiple of 2 (i.e., cross out 4, 6, 8, etc.—the rest of the even numbers). 3) 3) Circle the next unmarked number, 3. Then count by 3, crossing out each multiple of 3 that has not already been crossed out (i.e., 9, 15, 21, etc.). 4) 4) Continue in this manner, circling primes and crossing out composites, until you are done sieving.

Sieved-out Prime Numbers After you are done sieving, any numbers that remain (have not been crossed out) are prime numbers. Do you see any patterns in the Sieve of Eratosthenes?

Patterns One of the educational advantages of the Sieve of Eratosthenes is that it helps to develop your ability to see and extend patterns. Note the even numbers, for example. Do you see an actual pattern reflected in the table? What about multiples of 5?

Other ways to mark the prime and composite numbers: Circling and crossing off numbers is just one way to mark the prime and composite numbers in a Sieve of Eratosthenes. Some people color-code the numbers (e.g., marking the multiples of 2 in red, 3 in green, etc.), as in the earlier illustration. This helps them to find and analyze patterns. Can you think of other ways that you might mark the primes and composites?

Now that you’ve learned about the Sieve of Eratosthenes, do you see an error in this one shown to you earlier?

Just for fun (after trying the worksheet), check out one or more of the following sites and their interactive activities relating to primes and/or the Sieve of Eratosthenes: Sieve of Eratosthenes: Prime Number List Prime Factorization Machine

Sites to explore for more information on primes or Eratosthenes: The Prime Pages (University of Tennessee—Martin) Prime Numbers World’s Largest Known Prime Number Eratosthenes of Cyrene (University of St. Andrews) and.ac.uk/history/Mathematicians/Eratosthenes.html and.ac.uk/history/Mathematicians/Eratosthenes.html

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