Numbers in their prime
Q I am a newly qualified teacher and maths is not my specialist subject. I have been teaching prime numbers using the Eratosthenes sieve method. We made some pretty patterns but are there any other ways that would help pupils understand what is meant by a prime number? Why are prime numbers important? I was told there was a substantial prize for the next largest prime number: is this true? Are there mathematicians who research prime numbers? Any other facts about prime numbers that would interest my pupils?
A A great place to get information is the annual maths teaching conferences run by various associations. This year a combined conference with the title "Routes of Unity" will be held at the University of Warwick March 30 to April 2 2005 (details below).
An approach I have used successfully in teaching prime numbers has been through a structured investigation. The first part of the investigation is to make sure pupils understand what is meant by factors of numbers. Pupils work in pairs. Give each pair a set of 20 multilink cubes and a blank table to complete. Begin the session by explaining what is meant by factors of a number, a nice analogy is that factors are like "factories" - they are the numbers that multiply together to make another number. Show them an arrangement of six as a two-brick by three-brick rectangle (two and three are both factors of six). When they have found a pair of factors, record them in the table (a partially completed table is below).
I usually have some completed tables for the next stage so that pupils can check their own answers and have the correct information for the next part of the exercise. Each pupil is handed a blank graph, with numbers up to 20 to complete as shown. Instruct them to colour in the bars to represent how many factors each number has. One has only one factor, two has two factors and so on. If they mark those numbers that have the same number of factors in the same colour, the prime numbers can easily be identified.
Have a completed graph to put on the board or on an overhead projector slide for the class to discuss. What do they notice? This leads to the numbers that have only two factors - one, and the number itself - which is the definition of a prime number. One isn't a prime number as it only has one factor. Two is the only even prime number. Square numbers have an odd number of factors. Ask them to list the prime numbers separately (2, 3, 5, 7, 11, 13, 17, 19).
The largest prime number is 2 to the 24,036,583th power minus 1. It has 7,235,733 decimal digits and was found by Josh Findley of Orlando, Florida on May 28, 2004. This is the 41st Mersenne prime to be found (when 2n-1 is prime it is said to be a Mersenne prime).
One of the most important applications of prime numbers is in the encryption of data for finance, top secret ciphers and transactions on the internet. The Rivest-Shamir-Adleman (RSA) algorithm is the most widely used for the purpose. This uses public-key encryption. The security uses the fact that finding the prime factors of a large number is very difficult.
Prime numbers are also used in random number generation, eg the "random" function of a CD player.
Interesting fact: Goldbach's Conjecture (1742) asserts that every even number greater than four can be written as a sum of two primes, eg 10 = 7 + 3; 24 = 11 + 13. From 2000 to 2002, Faber and Faber offered a $1m prize for anyone who could prove the conjecture (the prize was unclaimed).
Prime numbers websites include: http:home.ecn.ab.cajsavardcryptopk0502.htm
www.mersenne.orgprime.htm (at this site they talk about the $100 000 prize to the first person or group to discover a ten-million-digit prime number) For details of Routes of Unity: www.bcme6.org
Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses.
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