Today, we are going to do some math -- unsurprising -- but we're going to take some things that have been in a Numberphile before and we're going to do a little bit more math with them. So, perfect numbers -- a bit of a staple of recreational mathematics -- everyone loves perfect numbers -- the smallest perfect number is six. [The] reason we call it a perfect number is because it equals the sum of its proper factors. So 6 is divisible by 1, is divisible by 2, it's divisible by 3, also divisible by 6? So I could put six on this list, but proper factors mean numbers that go into a number but aren't the number itself -- in fact, we can remove six from the list and six equals one plus two plus three, [this] equals the sum of its proper factors. The next number which this works for is 28, because that equals one plus two plus four plus seven plus 14, which are all of its -- all those proper factors summed together give you that number. You could -- some people like to say, perfect numbers are equal to -- like all of their factors equal twice [the number]. So if I did -- if I was super pedant, and instead of using proper factors I used all the factors, that would give me 56, which is twice because we are just adding the number on the end right, so yeah, same thing. I just -- I ignore that, and just stick with the proper factors. The next one Brady is? (Brady:) 496? (Matt:) 496! You've been in this game for too long alright. So you've memorized 496 and obviously I won't quiz you because obviously you know that it is one plus two plus four plus eight plus 16 plus 31 plus 62 plus 124 plus 248, there we go, all the proper factors of 496, if you add them all together you get 496 next one is 8128 right and so that -- and I won't list them all -- happens you get there, you add all those proper factors and equals that number as, it is -- carrying on -- and these were all the ones known back in the day, and we since found more of these perfect numbers, and they're wonderful. But the reason we found them is because of our friends, Mersenne primes, which of course [are] another staple of Numberphile So let's switch colors. So, Mersenne primes are 2 to the n minus 1, where that equals a prime number. So -- so that equals some prime because not all 2 to the n's minus ones give you a prime, in fact, let's run through a few very quickly. So if I come over here, these are going to be my values of n. These are going to be my 2 to the n's minus ones, As if I start when n is 1 so that's going to be 2 minus 1, that equals 1, is that prime? (Brady:) No. Well, there's a whole argument to be had there... Sometimes it is, sometimes it isn't -- generally speaking, no, that doesn't count, ok, let's do n is 2, ok. So that's 2 squared, that's 4, minus 1 is 3, that is prime, awesome. Ok next one up, 3, that gives us 7, that is prime, brilliant, ok 4, 4 gives us 15: not prime. That doesn't work, so as you can see as we go along, some of them work, 31, brilliant! That gets to stay, some of them don't -- 6, that gives us 63, no good, right, and so, some, but not all values of n [with] 2 [to the] n minus 1 give us a prime number out the other side, and some people like looking for patterns and we call those people "mathematicians" and so some people are like: "well, hang on, this is only working when n is prime." Didn't work for one, not prime, two and three work because they are prime, didn't work for four, not prime, prime, not prime, prime, and so on, and so in fact, if you try it for eight, it won't work, because that's 255, not prime, won't work for nine because that's 511, which looks very prime-y. That's got a real prime feel to it, not prime, in fact, when we check in 10 -- people get very -- I mean, 1023, I mean that looks like it's got a bit of prime going on that. It's not prime. Next one, 11, 11 is prime and over here is 2047, right, and that is not prime, not prime whatsoever and I think it was divisible by 23 or something. And so people look at that and go: "well that must be prime", because they see this pattern and think it must work -- doesn't hold, doesn't work for 11 in fact well, the next one it works for is not 12 because that's 4095, which we know is out of here, the next one is 13, which is 8191, that is prime, that one works, and so, in fact, that pattern, when people spot these prime values of N, giving these numbers prime, Only works in one direction as such okay? Let me -- let me be very clear with this -- one way we could describe this pattern is, if n is prime, then 2 to the n minus 1 is prime as well, but that doesn't work, we just proved that. But we haven't disproved the reverse, and the reverse of a statement is different, so for example I could say all -- all girls can wear hats. Logical statement, you're a girl, you can wear a hat, right? But I can't then say, if you can wear a hat, you must be a girl, right, it doesn't go the other way, right, because you know... (scrambling for words) Well the ability to wear a hat is necessary, but not sufficient to be a girl, anyway, the point is, right, the reverse is a different story, and in this case the reverse is true -- if 2 to the n minus 1 is prime, then n must have been prime. So the backwards still [holds], which is why it will still only be the primes, it just won't be all of them. So I guess these are our friends the Mersenne primes, these are our friends the perfect numbers, so far all very standard. But then suddenly you realize there's a link, if you look over here in our list before, three [is a] Mersenne prime, seven [is a] Mersenne prime, 31 [is a] Mersenne prime, in fact, we're picking them off in order, look down here 127 is the next prime and there is a factor of 8128 in fact, 8128 equals 127 times -- what's that going to be -- it's going to be 64... Yes, all even perfect numbers -- [I'm] being very careful here -- have a Mersenne prime factor. In fact, that's how we find perfect numbers, whenever a new perfect number is announced -- the most recent one was January 2013, the reason we found it, is someone has found a new Mersenne prime, because the great internet Mersenne prime search is chugging along -- every Mersenne prime has a corresponding perfect number and -- this is all just the introduction, by the way, this is the recommended -- you should have done this reading in advance, to be entirely honest. Well, [what] I'm going to show you is what that link is and prove that it always works. (Brady:) So now we start -- (Matt:) Now we start the video, so, are you ready to begin Brady? I'm gonna need some brown paper for this video and we will get this party started.
In the second part, where Matt proved it, I paused the video and tried to prove it myself, and this is what I came up with (which after viewing the proof, I can see similarities) - curious to see what people think:
We were given the case that:
(2n - 1) x (2n-1) = Perfect Number (P) for any case where (2n -1) is prime (ie a Mersenne Prime)
My proof is as follows:
Step 1:
(2n - 1) is equal to the sum of all the powers of 2 from 1 to 2n-1.
This can be proved a number of ways, however I just think of it in terms of binary numbers.
2n is written in binary as 1 followed by n zeros, for example: 32 = 25 = 100000 in binary.
If you subtract 1 from that number, you are left with n ones, so 11111, which is 1 + 2 + โฆ. + 2n-1
Step 2:
Our perfect number is the Mersenne Prime multiplied by the power of 2 just below it. We can then obtain the new factors simply by multiplying it by two, which corresponds to the previous power of two down the chain.
So with: P = (2n -1) * (2n-1)
It is also the case that: P = 2 * (2n -1) * (2n-2)
However, as we have proven that the Mersenne prime is equal to the sum of powers of two below it, that means another way of getting to that next factor is simply to add the Mersenne Prime to all of the powers of two below it.
This same logic is true every time you multiply the Mersenne Prime by two, you can simply add on the next number to the sequence to generate the next factor of P.
This will continue until you reach P (which will be 1 x P and thus the end of the chain). Because the Mersenne Prime is multiplied by a power of two to get to P, it means continually doubling it will eventually get to P.
Because doubling it is the same as adding itself to itself, and we proved in Step 1 that the Mersenne Prime is equal to the sum of all of the powers of 2 below it, we have proven that all of these numbers do indeed add up to the Perfect Number.
That just leaves us to prove we have found all of the factors
Step 3:
The only factors of a power of 2 are 2.
There are no factors to a prime number, so there are no factors to a Mersenne Prime.
That means the only remaining factors to find of the Perfect Number are the multiples of the Mersenne Prime, which all pair off each time you double the Mersenne Prime, its corresponding factor is the previous power of two. Until you get to 1 which pairs off with the Perfect Number itself.
There cannot be any more factors of the Perfect Number.
However this demonstrates why taking a 2n - 1 that is not prime does not yield a perfect number. For example 15.
Yes, 1 + 2 + 4 + 8 + 15 + 30 + 60 = 120 = 8 x 15
But 15 has additional factors of 3 and 5, meaning that these number also divide into 120.
What's the link to Objectivity?
Nevermind, found it: https://www.youtube.com/c/objectivityvideos
The nerd in me would love to see the other half of this proof. I seem to remember that not only are we guaranteed an even perfect number for every mersenne prime, but we can in fact show that every even perfect number MUST be a power of two times a mersenne prime.
Put more succinctly: if x is an even perfect number, then x=2n-1 (2n -1) where 2n -1 is a mersenne prime.
Of course I could just look it up, but I like these explanations.