Jenny Houssart examines the mystery, history and beauty of some famous numbers Illustrations by Bill Sanderson.
Mathematicians use lots of words in slightly different ways to the rest of the world; "prime" is a prime example! Miss Jean Brodie considered herself in her prime and Tony Blair is prime among ministers, but to mathematicians a prime number is a number with only two factors: itself and one.
We have known about prime numbers for thousands of years and many great mathematicians have wondered about them; some have even given their names to the theories or types of primes they have worked on.
The search for large prime numbers has exercised many great mathematicians and their computers. However, small numbers can be interesting too and the existence of 25 prime numbers under 100 means that investigation of primes is well within the grasp of many primary pupils.
The idea of prime numbers may only be explicitly introduced to children fairly late in their primary schooling, but it is the culmination of much prior work. In particular, it builds on the ideas of finding factors of numbers and looking at multiples. Work on multiples, factors and primes is particularly appropriate in the numeracy strategy, given its emphasis on oral and mental work. Knowing multiplication facts - and hence recognising multiples and factors - is a central activity.
Discussion of number properties and use of appropriate language are similarly recognised as important. So, for example, children might be shown a number and asked, "Is it a multiple of five? How do you know?" This can also lead to the idea of checking with one group of children asked to give, for instance, multiples of 10 and others asked to say whether the first group are correct and say how they knew.
Many children will learn to spot factors by becoming familiar with the patterns in multiplication tables. It can be a source of delight that they know that 2 is a factor of a large number such as 8,465,342, just by looking at the units digit and without doing any calculations.
Looking for factors other than 2, 5 and 10 requires a little more work, but in some cases there are divisibility tests which children can use. For example, you can check whether a number is a multiple of 9 by adding the digits to see if the answer is 9. So 45 is a multiple of 9 because 4+5=9, but 71 isn't because 7+1=8. For larger multiples of 9, such as 378, the process needs to be repeated. Adding the digits gives 3+7+8=18 and adding the digits in the answer gives 1+8=9. A similar test can be used to see whether three is a factor of a number, but in this case the digits will total 3, 6 or 9. As children become more adept at finding factors of numbers, they will be able to identify those which have no factors except themselves and 1, and their study of primes can begin.
'In your prime...' There is more to studying prime numbers than identifying factors. Investigations involving prime numbers are ideal for developing mathematical reasoning. Many such problems require the testing of examples, and children will soon realise that eliminating whole categories of numbers is much faster than testing every case. For example, they will quickly see that even numbers above two need not be considered when looking for primes. This provides a good opportunity for them to explain their reasoning.
Testing for primes also involves another important mathematical idea - that certain properties of numbers are necessary, but not sufficient, to make them primes. For example, if a number with two or more digits is prime, it must end in 1, 3, 7 or 9. However, ending in one of these digits does not mean that it is prime, simply that it might be. Thus, you can tell by looking at the final digit that 596 is not prime. The same process tells you that 583 and 587 might be prime, but further investigation is needed to confirm this. In fact 587 turns out to be prime, but 583 is 53 x 11.
An advantage of studying primes is that it can be made more or less challenging not only by the amount of guidance given, but by the size and range of numbers studied. So children may be asked to search for prime numbers where all the digits are also prime. If the problem is restricted to two-digit numbers and you have already established that the prime digits are 2, 3, 5 and 7, then the problem is quite manageable. Giving less guidance and specifying more digits increases the challenge. Leaving the problem completely open raises the possibility of an infinite number of solutions.
Children can be interested in the existence of large primes and amused by their names. For example, a "titanic prime" has at least 1,000 digits; a "gigantic prime" has over 10,000. Your pupils may not reach such heights, but by studying primes they will learn something about properties of numbers and may see their beauty and mystery.
Finding primes to 100 There are 25 prime numbers under 100. The method of finding them, outlined below, is known as Eratosthenes' Sieve (see 100 square below and key):
* Cross out 1 which is not a prime.
* Cross out multiples of 2 other than 2 (shown in red).
* Repeat for multiples of 3 (green), 5 (blue) and 7 (orange).
Watch out for ...
In looking for primes up to 100, you only need to eliminate multiples of 2, 3, 5 and 7. There are two reasons why you do not have to consider multiples of other numbers:
* You don't need to test for factors above 10. If a number below 100 has a pair of factors, one of them is less than 10. For example, 51 is not prime because it is 17 x 3. It has already been crossed out as a multiple of 3, so it doesn't matter that you haven't tested for multiples of 17.
* You don't need to test non-prime numbers. For example, all multiples of 4 have already been crossed out because they are multiples of 2.
Similar rules apply when testing any number to see if it is prime. When you are testing for factors you only have to try prime numbers up to the square root of the number you are investigating.
Using Calculators This certainly makes sense if you are testing large numbers, but it does require understanding and a systematic approach.
Children will need to understand that you can test for factors by dividing and seeing whether you get a decimal or whole number answer.
They will also be helped by understanding which numbers they have to consider when looking for factors. (See the "Watch out forI" section above.) The process will also be quicker if they eliminate some numbers using non-calculator methods. For example; is 743 prime? Using a calculator gives the square root of 743 as 27.258026.
This means we only have to test primes up to 27. These can be listed as 2, 3, 5, 7, 11, 13, 17, 19, 23. You can tell by looking at the final digit that neither 2 nor 5 are factors of 743. Three can be eliminated by adding the digits. The other six numbers can be divided into 743 in turn using a calculator.
None of these give whole-number answers and therefore 743 is prime.
Investigating Primes The following investigations are truly "open-ended" as most have an infinite number of solutions. All can be presented in a more "closed" fashion by restricting the size or range of numbers searched for. Look for:
* prime numbers, which are also palindromic (the same backwards);
* prime numbers in which all the digits are also prime;
* primes, which if you double them and add one, give another prime (these are known as Sophie Germain primes. Repeat the process and try to make a "chain" of primes, where each is double the number that comes before it, plus 1);
* pairs of prime numbers, which are only two apart (twin primes).
Investigating Primes - the answers Answers are given up to 1,000 for the first two problems and up to 100 for the next two. All the problems have further solutions.
* Palindromic primes (up to 1,000): 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929.
* Primes with prime digits (up to 1,000): 23, 37, 53, 73, 223, 227, 233, 257, 277, 337, 353, 373, 523, 557, 577, 727, 733, 757, 773.
* Sophie Germain primes (up to 100, with resulting chains): 2, 3, 5, 11, 23, 29, 41, 53, 83, 89 chains: 2-5-11-23-47, 41-83-167, 89-179-359-719-1439-2879.
* Twin primes (up to 100): 3,5 5,7 11,13 17,19, 29,31 41,43 59,61, 71,73.
Further Information * Numbers: Facts, Figures and Fiction by Richard Phillips, Cambridge University Press. This is an attractive book for classroom use that deals with primes along with other properties of numbers.
* Further information on primes and related ideas such as divisibility tests can be found in the "Did you know?" section of the Maths Year 2000 website at www.mathsyear2000.orgdidindex.html.
* Information on the life and work of Eratosthenes and Sophie Germain can be found in the biographies section of the St Andrew's University history of mathematics site at www-history.mcs.st-and.ac.ukhistory.
* A multiple of a number is obtained by multiplying it by any whole number. For example 3, 6, 9 and 12 are multiples of 3.
* A factor of a number (sometimes called a divisor) divides exactly into it. For example, 2 and 4 are factors of eight. One and 8 are also factors of 8, because 1 is a factor of every number, and every number is a factor of itself.
* A divisibility test is a way of telling whether a number has a particular factor or divisor. For example, you can test whether 10 is a factor of a number by seeing whether the units digit is 0. You can test for divisibility by 9 by adding the digits of a number and seeing whether the result is 9.
* A prime number has exactly two factors: itself and 1. For example, 5 is prime as it cannot be obtained by multiplying two smaller whole numbers. So 6 is not prime because it has four factors: 1, 2, 3, 6. And 1 is not prime as it has only one factor - itself.
Jenny Houssart is research fellow at Centre for Mathematics Education at the Open University