What are prime numbers?
Prime numbers are the building blocks to all integers. A prime number is any integer greater than 1 that is only divisible by 1 and itself. The integer, 1, is niether prime nor composite.
What is the use of prime numbers?
We can use prime numbers for cryptography. Cryptography being hidden messages in which only people who are intended to read it can. Prime numbers can be found in music as well.
How do we find prime numbers?
The Sieve of Eratosthenes is a method to find prime numbers. Eratosthenes of Cyrene (276-294 BCE), created this method where the composite numbers will "fall through" the "sieve" and prime numbers will be collected. First make a table of integers, 1 - x, where x is an integer greater than 1 and the integers are in numerical order. Since 1 is not prime, begin by crossing off it in your
table. We will circle 2 since it is the first prime number because the only factors of 2 are 1 & 2. Now we will cross out all multiples of 2 since multiples of that particular integer are at least 1, 2, and itself. Also, we may note all even numbers, are not prime numbers, with the exception of 2. The next number not circled is 3, and the mulitples of 3 are only 1 & 3, therefore being our next prime number. Now cross out multiples of 3. The next number which has not been crossed out is 5, which is the next prime. Repeat the process.
Primes in the classroom
Using the Sieve of Eratosthenes, we may uncover concepts and patterns with prime numbers, following a discovery method to gain conceptual understanding of the primes. My favorite way to teach mathematics is to use the discovery method, where students must use problem solving methods to find patterns and make conjectures. Using the method above students will use the sieve to "collect the primes." Then follow guided & purposeful questioning to uncover secrets about prime numbers. Such as:
Why is 2, an even integer, prime?
Where can we stop looking for multiples?
Did you know?
The largest prime number 17,425,170 digits long and was found in 2013.
In 1742, Christian Goldbach conjectured that every even number (except 2) can be expressed as the sum of 2 prime numbers. Can you find some?
The above conjecture is unproven!!
References:
“Sieve of Eratosthenes.” SIEVE OF ERATOSTHENES (Java, C ) | Algorithms and Data Structures, www.algolist.net/Algorithms/Number_theoretic/Sieve_of_Eratosthenes.
“Student Explorations in Mathematics - Unlocking Primes.” Nctm.org, 3 Sept. 2015, www.nctm.org/Publications/Student-Explorations-in-Mathematics/2015/Unlocking-Primes-_September-2015_/.
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