8 An unsolved problem!
A lot of the mathematics you have looked at so far in the course has been used to help people solve problems for centuries. However there is much more to mathematics than that! Exciting and new developments are being made all the time and there are many problems that mathematicians have not managed to work out. This section takes a brief look at one of these problems.
It was unsolved at the time of writing!
In 1742, a Prussian mathematician, Christian Goldbach, made the following conjecture:
All positive even integers bigger than or equal to 4 can be expressed as the sum of two primes.
A conjecture is a suggestion that has not yet been proved to be true.
This statement contains quite a lot of mathematical terminology, so before going further, here are some definitions.
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An integer is a whole number, negative, zero or positive; for example, −6, 23, 0 and 281 are all integers.
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An even integer is one that can be divided exactly by two; for example, −8, 0, 42, 128 or 1002. The last digit of an even integer is always 0, 2, 4, 6 or 8. Integers that are not even are odd.
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A prime number is one that can only be divided exactly by 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59.
Now, break down Goldbach's conjecture into steps.
A lot of encryption processes depend on prime numbers.
‘All positive even integers bigger than or equal to 4 …’ means the numbers 4, 6, 8, 10, 12, and so on.
‘… can be expressed as the sum of two primes’ means that you can choose any of the numbers from the prime number list to add together. For example, 8 = 5 + 3.
Activity 25: Prime numbers
Try to express the following numbers as a sum of two primes.
4 12 28 40 62
For each number, try to find as many combinations of two primes as you can.
Do you think Goldbach's conjecture is true?
Discussion
It can help if you look at the list of prime numbers systematically, starting with 2. Is there a number in the list that you can add on to 2 to get the number you want? Then look at 3, and so on. You should be able to find the following combinations:
4 = 2 + 2;
12 = 5 + 7;
28 = 5 + 23 or 28 = 11 + 17;
40 = 3 + 37 or 40 = 11 + 29 or 40 = 17 + 23;
62 = 3 + 59 or 62 = 19 + 43 or 62 = 31 + 31.
Are you convinced that Goldbach's conjecture is always true?
Although the conjecture does work for the few numbers we have tried here, that does not necessarily mean that it is going to work for all the even integers greater than or equal to four. The even integers go on for ever, so it is not possible to check them all individually either by hand or computer – some other way of proving the result is needed. Although higher branches of mathematics have been successful in tackling a lot of similar problems, this one is turning out to be particularly tough. So tough in fact that, in March 2000, the publishers Faber and Faber offered a $1 000 000 prize to anyone who managed to prove the conjecture by March 2002 and no one did! So even though many great mathematicians have attempted a proof, over 250 years later, it still hasn't been sorted out. You might find that quite reassuring the next time you get stuck on a mathematics problem – being stuck really is a natural state for a mathematician! It's often the time when great discoveries are made or when the best learning takes place.
There is also a $1 000 000 prize on offer for each of the seven great unsolved problems of mathematics known as the Millennium Problems.