Great Uncle Fred? We were doing prime numbers and it was interesting but the lesson finished and I had missed something and next lesson it’s something else. The teacher was explaining that the biggest known prime number is an extremely big number but he said that there cannot be a biggest prime number because... And I didn’t catch that bit and that’s my question - how does he know that we will never know what the biggest prime number is? (Robert, Spring 2008)

Robert, to put your question another way - ‘How can you prove that there is an infinite number of prime numbers?’ Well, I am not a mathematician at all, but I happen to know that this question was answered by a gentleman called Euclid in about 300 BC! As you know, a prime number is a number that can be divided only by itself or by one. The only even prime number is two, of course, because all other even numbers can be divided by two. There must be an infinite number of prime numbers because... ...all the prime numbers multiplied together, plus one, would have to be another prime number! You can see this with just the first few prime numbers: 2 x 3 = 6 6 + 1 = 7 (a prime number) 2 x 3 x 5 = 30 30 + 1 = 31 (a prime number) 2 x 3 x 5 x 7 = 210 210 + 1 = 211 (a prime number) 2 x 3 x 5 x 7 x 11 = 2310 2310 + 1 = 2311 (a prime number) It’s actually very simple. The multiple of any number, plus one, cannot be divided by that number. Thus is proved the existence of infinity. As you can see from those first small sums, the figures mount up very quickly. It’s not long before you’re trying to think about absolutely enormous numbers. So far, as your teacher might have told you, the biggest prime number to have been worked out is 9,808,358 digits long. Can you imagine that? When you consider that the number one billion is ten digits long? Of course, if they want to find an even bigger number, they need just take that number and all the other prime numbers smaller than that number, multiply them together, and add one. But I think they might need a bigger piece of paper. As I say, I am not a mathematician but I agree that there is something fascinating about prime numbers, not least this relation to infinity. They are useful in countless other ways too - they can help you find your way to the centre of any maze, apparently - but the limits of my knowledge are fast approaching. Good luck!

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