Marcus Du Sautoy, professor of mathematics at the University of Oxford, recently said that he would like to teach all the world that the number of prime numbers is infinite. So I decided to ask a Year 11 class how many primes there are.
One pupil said, slightly puzzled: "I mean, they've got to stop somewhere, haven't they?" Well, they certainly seem to get more spread out as our primes get bigger. We can have strings of whole numbers that are as long as we like yet don't feature a single prime: consider the numbers from n! + 2 to n! + n, none of which can be prime. But do they ever peter out completely?
The Greek mathematician Euclid gave us a proof of timeless simplicity 2,000 years ago that the primes never stop. Our pupils, however, may need a little preparation to fully appreciate its beauty. One fact is necessary: that all numbers can be broken down into primes. Once your pupils are happy with this, try the following: "Give me a whole number bigger than 1 that gives a remainder of 1 when we divide by 3 or 5 or 7."
The smallest example is 106: how can we find this without a lot of guessing? Well, 106 = 3 x 5 x 7 + 1, and once our pupils have understood the construction here, the sky is the limit. "Give me a whole number bigger than 1 that leaves a remainder of 1 when divided by 23 or 29 or 31." We can immediately say the smallest example is 20,678, as given by 23 x 29 x 31 + 1.
Now we can go a step further. "I'm going to show you how, given a list of primes, we can always find another one." Suppose our list to start with is 2, 3, 7, 11. Now consider the number 2 x 3 x 7 x 11 + 1. What is the remainder when we divide this number by 2? Clearly it is 1, and likewise for dividing by 3 and 7 and 11. In fact, the number is 463, which is prime, and so we have found a new prime for our list.
Will this always work? Let's try 2 x 5 x 7 x 11 + 1. This is 771 = 3 x 257. So this time the number we make is not prime, but it is a product of primes, none of which are in our initial list. So once again, we have found a way to extend our list of primes.
Now we are ready to head down Euclid's road. Suppose the list of all primes is finite, so we can write them down: 2, 3, 5, 7... pn. Now let's make the number 2 x 3 x 5 x 7... x pn + 1. Certainly none of the primes in our list go into this - they all give remainder 1. So this number could be prime, in which case we have a new prime to add to our list. Or it is the product of primes that are not in our list, so they are new too. So we have a contradiction (not allowed in maths) and our initial thought that the list of primes is finite must be wrong. Delicious.
Jonny Griffiths teaches maths at a sixth-form college
Test pupils' knowledge of prime numbers with a true or false quiz by jog_on.
For some musical maths try Mr D's Transformers-themed prime number song.
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