Times Tables and Goldbach's Conjecture |
Christian Goldbach was a mathematician who lived in Prussia, which is now part of Germany, from 1690 to 1764. In about 1740 he came up with the idea that: Every even number greater than 2 can be written as the sum of two prime numbers. I can hear you saying it: “It’s those times tables again….” This is now known as Goldbach’s Conjecture and you can test it for yourself, but first I need to make sure you understand what it means. When we say ‘the sum of two prime numbers’, we mean two prime numbers added together, so what Goldbach was saying is that you can always find two prime numbers that will add up to any even number you want to choose that is greater than 2. Obviously the first one is therefore 4. It is an amazing idea because he is saying that you can find two prime numbers that will add up to any even number, even if that even number has billions and billions of digits. There is no way that he could have checked this by hand. In fact, even with today’s super fast computers, it has only been checked up to about a trillion (1 000 000 000). It is called a ‘Conjecture’ which means a very good idea that looks as though it might be true, but no-one has proved that it definitely is true for all the even numbers greater than 2. Even today, no brilliant mathematician has managed to prove it is true for all even numbers, although no-one has found an even number that does not work! Mathematicians are very clever people, of course, and they have ways of proving things for all numbers of a certain type without having to test them all on computers, but so far they have not managed to do this for Goldbach’s Conjecture. Unfortunately, I can’t explain how they do this here. So, let’s get started. First of all you will need a list of prime numbers, of course, and by now you should be able to write the first few down from your head and calculate the next few quite easily using your knowledge of factors and multiplication tables. You will find testing Goldbach’s Conjecture very difficult without this list, so begin by writing down as many prime numbers as you can. I shall give you the first few to get you started. 2, 3, 5, 7, 11, 13, 17, 19, 23 … We will begin at the beginning with the smallest number we are considering, which is 4. 4 = 2 + 2 and straight away you will see that we need to use the same prime number twice. This is allowed. 6 = 3 + 3 Right, off you go. How far can you get? No calculators, of course.
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