What do the numbers 6, 28, 496 & 8128 all have in common? Aside from being even numbers they are also all examples of perfect numbers.
To see what this means take a look at the factors of the number 6 (the factors of a number are those whole numbers which divide into the original number exactly). The factors of 6 are 1, 2, 3 & 6 (because 6 can be divided by these numbers exactly).
A perfect six!
Now ignore 6 and add the other factors together and you get 1 + 2 + 3 = 6 which is the number we started with – this is what makes 6 a perfect number. Likewise the factors of 28 are 1, 2, 4, 7, 14 & 28 and if we add these factors together (ignoring 28) we get 1 + 2 + 4 + 7 + 14 = 28 and so 28 is also a perfect number. Hence a perfect number is any number which is the sum of all its factors other than itself.
Perfect numbers are pretty rare and you can easily verify for yourself that most numbers are not perfect – for instance the number 8 has factors 1, 2, 4 & 8 and 1 + 2 + 4 = 7. In fact they are very rare and at the last count only 44 perfect numbers have ever been found – the fifth perfect number (after 6, 28, 496 & 8128) is 33,550,336 and the largest known one has nearly 20 million digits!
Extra special 28!
So far no-one has ever found an odd perfect number – it seems pretty likely that there are no odd perfect numbers but this has yet to be proved and so perhaps there is one or more out there to be found; it would be a good way to become famous in the field of mathematics if you could find an odd perfect number! We also don’t know how many perfect numbers there are. As I’ve mentioned, only 44 have been found so far but new ones are still being found fairly regularly and it’s quite likely that there are infinitely many out there to be discovered some day but again we can’t say for certain – this would be another guaranteed way to make your name in maths if you could prove that there were infinitely many perfect numbers!
Perfect numbers exhibit several other interesting properties. For example, if you reciprocate the factors of a perfect number and add the answers together you will always end up with the number 2 (if you reciprocate a number you divide 1 by that number, for example if you reciprocate 3 you get 1/3 which we call the reciprocal of 3; similarly the reciprocal of 7 is 1/7). For example, we can add together the reciprocals of the factors of 6 and we get 1/1 + 1/2 + 1/3 + 1/6 = 2.
496 - in the perfect club!
To be honest, perfect numbers are not particularly important numbers but that’s not really the point – if people find them interesting, and indeed fascinating, as many mathematicians have over the years, then who needs any other reason to take an interest in them? Personally I’ve always found them to be a source of amusement and fun and I hope other people do too!
Find out more about studying Maths at Bellerbys College by visiting the Bellerbys courses page.
Tags: Perfect numbers
October 19, 2009 at 10:38 pm |
youre cool