Mathematicians like to prove things. Things that most of us would take for granted, statements like 1+1=2 are of no use to a mathematician unless it they can be ‘proved’ – demonstrated to be true in every circumstance. In Bertrand Russell and Alfred North Wallace’s great work, The Principles of Mathematics, several hundred pages are devoted to proving that 1+1=2. Why? Because in other places mathematicians will want to use the statement that 1+1=2 as part of the proof of something far more complex, and any proof is only as good as the parts that make it up. The history of mathematics is littered with proofs that relied on something that was ‘self-evident’, only to have it turn out years later that the proof is useless because the self-evident something turned out not always to be true.
It’s every mathematician’s dream to produce a proof of something that has never been proved before (or to demolish someone else’s proof). Many of the propositions that have been put forward but not yet proved are exceedingly complex but some are so simple that you’d think it would take no more than a few minutes to work out why they ‘must’ be true.
So it was that back in the 18th century, a German mathematician named Christian Goldbach suggested that every even number other than 2 could be produced by adding together two prime numbers – numbers that can only be divided by themselves and 1 without leaving a remainder. It doesn’t take long to demonstrate that it is true for even numbers of reasonable size and, in the centuries following, Goldbach’s ‘conjecture’ was found to be true of bigger and bigger even numbers. The advent of computers has led to the analysis of numbers of ridiculous size being considered and, yes, every one of them can be produced by adding together two prime numbers.
But no-one has ever proved that it will be true for every conceivable number, and there are plenty of cases of statements that turned out to be true over and over again, with bigger and bigger numbers, only for an exception to be found – often after many years.
So two centuries later Goldbach’s Conjecture remains just that, a conjecture. In practice, unless you’re going to be working with numbers that have 20 or more digits, you needn’t worry, because that’s how far the conjecture has been shown to be true. But don’t try telling a mathematician that the conjecture ‘must’ be true after all this time – or they may ask you to prove it!
Anyway this little illustration is hardly the most visually exciting of the collection but it does make the point – for the first few even numbers at least – and there are some interesting patterns if you look. Simply follow the lines from any pair of prime numbers, one on each side, and where they meet they are added together to produce an even number.
Before you ask, the number 1 isn’t included since mathematicians don’t consider it a prime number – you can work out for yourself why if you look back at the definition of a prime number given above.
And finally, a little problem to ponder. If the Goldbach Conjecture is ever proved to be true, it will also prove that any number is actually the sum of three prime numbers. Can you see how?