The pages on factors introduce the idea of perfect numbers. One way of defining these is to say that the sub-factors, all the factors except the number itself, add up to the number.
So fac(6)={1,2,3,6} so the sub-factorial sum is 1+2+3 = 6
similarly for 28 1+2+4+7+14 = 28
and the next one is 16x31.
and then …?
I see that 1+2 = 3 for 6 and that 1+2+4 = 7 for 28…..How about the factors starting 1,2,5 then 8, so the number to test is 40? This fails, and 16 is not a factor of 40. Nor does 80 work, since that includes the multiples of ten, And two.
Could it be that we should start by adding powers of two? 1+2+4+8=15, 8x15=120 but 120 isn’t perfect any more than 40 is. Why is that? Clearly, if you look at the case, because 120 has factors including 10 and 12, not in the generated list {1,2,4,8}. What about the next one, then, which would be 1+2+4+8+16=31, so 16x31 =496. Success; we showed in Factors that 496 is indeed perfect.
So, our ‘rule’ seems to be that we start with factors of 2: 1,2,4,8,16…. And we can use the sums of these, 3,7,15, 31 but not the 15. What is in any way different about 15 in that list? Is it that 15 is the only non-prime?
Let's have longer list, then:
Are any of the left-hand column not prime? I recognise 15,63, 255,1023 as having factors of 3 and/or 5, so I’ll take those lines out. Remember these powers of two add up to the number on the left. We are testing to see if the last power of two and the lefthand number have, as their product, a perfect number.
Are these perfect? If so, here’s an idea to explore: Look at the row below that starts 16x31. I’ve added a column that shows 16 = 2⁴ and that 31 = 2⁵-1
BUT, those bottom two lines don’t produce perfect numbers, though the NEXT line is perfect. Why is that? I think the answer is now obvious and you can now attempt to write the next few perfect numbers. There is only one small problem, that you probably can’t do this on your caculator or computer, as the numbers are too long. if 8128 is the 4th, then the 10th has 54 digits and the 20th has 2663. So you’re doing very well to write the eighth one down. I’ll tell you it looks a bit like 230……..128, but that’s quite enough help. The corrected column titled q, when you've written it, will become known as Mersenne primes.
I took several Y8 classes (and the occasional Y12) through this thinking, once they’d become familiar with the idea of investigations. Most of them spotted the powers of two and the primes without further prompting, but that was probably because we’d already spent so much time on powers of two and on primes.
Looking up the answers is cheating; all the excitement lies in having an idea and testing it out to see if it works or only works a bit. This is a good example of such a process; 8x15 = 120 ought to work but doesn’t, so why is that? 64x127 works but 256x511 doesn’t, so what is special about 64x127 that doesn’t apply to 256x511? Does that predict that we think 1024x2047 will fail, and what do we need to do to show that it will? Questions like this make maths as a subject have some meaning, as this is the same process that any sort of engineer will use when testing a process or a piece of equipment (civil, mechanical, electrical, biological) — because it is a process of investigation. The cycle is idea → test → modify idea → test and go round and round until either enough is found to explain what is going on (as seen so far) or the idea is rejected as non-workable. That experience usually teaches enough to make a better suggestion.
To me, this is the whole point of maths. Being given the answer makes it very hard to find new topics to explore. Finding the answer by looking at the work of others does the same — it prevents you from that same discovery. On this page, by having already shown that 496 is perfect, then the testing to see if 8128 is perfect is already that step more difficult. If you’d already discovered that I’ve told you 8128 is perfect without testing that for yourself, then the next one is a LOT of work and many people simply won’t bother testing it for themselves, saying it is too much work (which isn’t true, but it feels that way when you see the number for the first time).
What’s the point? I hope I’ve answered that. Who wants to know about perfect numbers? Very few people, though there is a big deal in discovering large primes for security and password purposes. However, the process is the thing worth exploring. My classes were surprisingly excited, though in each class was the odd one or two who wouldn’t engage. They would say everyone else was being odd, while accepting that they were suddenly in a world of nerds, excited by these stupid number things. I still say this was their loss and I am still sad at my failure to persuade them to engage.
DJS 20181004
Could there be any odd perfect numbers? This stays unproved when writing in 2018. That is, no-one has yet found one (that would be exciting, since it would generate a load more theory) nor is there yet a proof that there are none.
Just because you have a way of generating Mersenne primes does not mean you have found all primes at all. One counter example, a prime between 31 and 127 would do. How about 101,103,107,109, which are all prime? However, knowing some large primes means you can generate some very large numbers that you know will be prime. If p is prime then 2^p -1 is prime (shown by Euclid, way back, around 300 BC). So if q=2p-1 and r = 2q-1 and s=2r-1 you can make some very large numbers that you know will be prime. One such is 2^77232917 -1 which has over 23 million digits and is the 50th Mersenne prime. Link to why you might want to do this. I didn’t find this at all convincing, as if the big reason was being ignored, like the elephant in the room. The short answer: you would want large primes for security coding. Link to large Mersenne primes.
I have long thought that you could use any irrational number (an infinite non-repeating decimal) for a security code; if I needed a ten digit code, then I pick my n√p (say 19√82, if you're born in 1982) and only reference where the first non-zero digit is (perhaps today's day and month (1004 above), offset by a known but secret amount). Personally, I'd need a look-up table, but how one hides things like that, or tests a security code, is quite another game. Practically, you don't want any methods or classifications published for all to see as it makes it very much simpler to crack a code. But you might notice that you need two systems, Sender and Receiver, to agree that their codes match enough to open communications. So you might want for Sender to send say characters 33-42 (ten digits) and for the return check to be chars 43-52, which you can then do as often as seems necessary.
Car security keys work, we are told, on a file of "random" numbers held by the key fob and the car; each time a code has been used it is discarded, so that some sort of pointer to 'next number' is required. If that list of random numbers is fixed, and worse, knowable to others, it might as well not be random (it could just as usefully be telephone numbers, or any list at all. But for any system there needs to be a way or ways to adjust it, to modify or correct it and these are the weaknesses that any hacker will seek to exploit. For a converse viewpoint, consider cryptocurrency security, which is so good that there is an awful lot of alleged value locked behind inaccessible codes, which have been lost for any of many reasons. Such as, the hard drive where I'd put the very long passcode has been wiped. In effect, any such secure code is valued at the size of the account. Small things are very easily lost.
What is really required for a secure key is a way of showing that two things (key and lock) are paired without easy duplication. Just what you mean by security and how hard you want to work for that lack of duplication are consequences.
