Season 10 Episode 4
On this episode, Zoë tells us about multi-perfect numbers; they're even better than perfect numbers!
Further Reading
Make your own Multiperfect number
Here’s an article that describes the process we were following during the episode. A DIY Project: Construct Your Own Multiply Perfect Number! by
Seth Colbert-Pollack, Judy Holdener, Emily Rachfal, and Yanqi Xu.
We didn’t show this in the OOMC episode, but it’s possible to get “stuck”, if you want to cancel a prime factor from the numerator but you’ve already included it (because modifying that factor to \(p^2\) would totally change the factor in the numerator and also change your decisions from there). An idea might be to code this up for a computer, instructing the computer to backtrack and try different things if it gets stuck. It’s not clear to me whether such an algorithm will terminate.
Multiperfect numbers
There’s some information about multiperfect numbers on Wolfram MathWorld. Note that the last paragraph has a typo; it starts “If \(n\) is a \(P_5\) number”, but probably means “If \(n\) is a \(P_3\) number”. We mentioned this fact in the episode; if your number is 3-perfect and not a multiple of 3, then multiplying it by 3 will give you a 4-perfect number. You might like to check your understanding by proving that to yourself. Then read the other fact at the end of that Wolfram MathWorld article about \(P_{4k}\) numbers.
Reading mathematics is often like this; you need to keep pausing and trying things for yourself, otherwise you’ll never understand it (and you’ll never spot any of the typos!).
Even perfect numbers
There’s a proof that all even perfect numbers (maybe I should say even 2-perfect numbers) have a particular form. You can do half of the proof yourself now!
Exercise: Suppose that $2^n-1$ is prime for some whole number $n$, and write $N=2^{n-1}(2^n-1)$. Show that the sum of the factors of $N$ (including $N$) is $2N$.
For the other half of the proof, due to Euler, see this webpage.
Odd perfect numbers
No-one knows any examples of odd perfect numbers, and no-one knows if there are any odd perfect numbers waiting to be found. We don’t know if there are infinitely many, finitely many, or none.
But we do know some things! Wikipedia has some facts about odd perfect numbers.
One of those facts is that if there is an odd perfect number, then it’s not a multiple of 105. This has been known for quite a while, and there's a nice write-up of one proof on the Bilkent University’s website after it was used for their Math Problem of the Month September 2006.
There’s more information about odd perfect numbers on Wolfram MathWorld. It’s one of those things where mathematicians have checked all the numbers up to \(10^{1500}\) without finding any, but we still haven’t proved there aren’t any. But if you're just looking for one, then we've seen that you don’t need to check every single number...
If you want to get in touch with us about any of the mathematics in the video or the further reading, feel free to email us on oomc [at] maths.ox.ac.uk.