There’s a fun mathematics video event called Clopen Mic Night and the December edition is available, for a limited time, for viewing on YouTube . One of the “acts” was Luna talking about multiperfect numbers — afterward I did some exploring of it myself. And I found this number: 518,666,803,200. (Not that I was the first to find it, of course, but I did find it on my own.)
A perfect number sometimes is defined as a number whose aliquot factors — the ones not including the number itself — sum to that number. For instance:
$l \displaystyle 1+2+3 = 6 $
where 1, 2, and 3 are the divisors of 6, other than 6. An equivalent definition is: A number all of whose factors sum to twice that number:
$l \displaystyle 1+2+3+6 = 2\times 6 $
or yet another way: A number for which the ratio of the sum of its factors to the number itself is equal to 2:
$l \displaystyle \frac{1+2+3+6}{6} = 2 $.
By contrast, most numbers have a non whole number for that factors-to-number ratio, for example:
$l \displaystyle \frac{1+2+3+4+6+12}{12} = 2.3333\ldots $ for 12.
Of course, for 1 the ratio is
$l \displaystyle \frac{1}{1} = 1 $.
And that’s the only number with a factors-to-number ratio of 1. But can this ratio be some other whole number? Like 3? Yes it can, and the smallest example is 120:
$l \displaystyle \frac{1+2+3+4+5+6+8+10+12+15+20+24+30+40+60+120}{120} = 3$.
Such numbers are called multiperfect numbers. The smallest examples with ratios of 4 and 5 are higher, and you might think it’d be pretty tedious to find them without a computer. But not so much!
Take a look at the factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120. Notice that half of them are multiples of 5 and the other half aren’t: 5, 10, 15, 20, 30, 40, 60, 120 versus 1, 2, 3, 4, 6, 8, 12, 24. And in fact the ones that are are 5 times the ones that aren’t. That means we can write:
$l \displaystyle 1+2+3+4+5+6+8+10+12+15+20+24+30+40+60+120 = (1+5)(1+2+3+4+6+8+12+24) $.
Now looking at the numbers in the second set of parentheses, half are divisible by 3 (3, 6, 12, 24) and half aren’t (1, 2, 4, 8), and the ones that are are 3 times the ones that aren’t. So
$l \displaystyle 1+2+3+4+5+6+8+10+12+15+20+24+30+40+60+120 = (1+5)(1+3)(1+2+4+8) $.
Now, the prime factorization of 120 is $l 2^3\times 3\times 5$. It’s a product of three prime powers: $l 2^3$, $l 3^1$, and $l 5^1$.
For a prime power $l p^n$, the sum of the factors is $l 1+p+p^2+…+p^n$. The factor sum for 5 is $l 1+5$, for 3 is $l 1+3$, and for $l 2^3$ is $l 1+2+2^2+2^3 = 1+2+4+8 $. So the factor sum for 120 is the product of the factor sums for each of the prime powers in its factorization.
And that works generally. Check for yourself, for instance $l 850 = 2\times 5^2\times 17$: $l (1+2)\times(1+5+25)\times(1+17) = 1674$, and the summed factors of 850 is $l 1+2+5+10+17+25+34+50+85+170+425+850 = 1674$.
And of course the factors-to-number ratio is the product of the factors-to-number ratios of the prime powers.
Notice some things: For a power of $l 2$, $l 2^n$, the factor sum is $l 1+2+…+2^n = 2^{n+1}-1$.
For a prime number $l p$, the factor sum is $l 1+p$.
If $l p$ is a Mersenne prime, it’s of the form $l 2^k-1$ and the factors-to-number ratio is $l \frac{2^k}{2^k-1}$. But the factors-to-number ratio of $l 2^{k-1}$ is $l \frac{2^k-1}{2^{k-1}}$. That means the factors-to-number ratio of $l 2^{k-1}\times (2^k-1)$ is $l \frac{2^k-1}{2^{k-1}}\times\frac{2^k}{2^k-1} = 2$. So if $l 2^k-1$ is prime, there is a corresponding perfect number, $l 2^{k-1}\times (2^k-1)$.
For instance: $l 2^3-1 = 7$ is prime. So $l 2^2\times (2^3-1) = 4\times 7 = 28$ should be a perfect number, and it is ($l 1+2+4+7+14+28=56=2\times 28$).
(You can also prove if $l n$ is an even perfect number, there is a corresponding Mersenne prime. And no odd perfect numbers are known to exist. So the known Mersenne primes and perfect numbers are in one to one correspondence.)
More generally, if you can find a string of factors-to-number ratios whose product $l \frac{a}{b}\times\frac{c}{d}\times\frac{e}{f}\times\ldots $ is a whole number $l n$, then $l b\times d\times f\ldots $ is a multiperfect number of order $l n$. (This requires $l b$, $l d$, $l f\ldots$ to be powers of distinct primes.)
For instance:
The ratio for 32 is $l \frac{63}{32}$. To get a whole number we need to cancel with the 32 in the denominator. But we also want to cancel whatever is in the denominator of the next ratio, so let’s use 7: its ratio is $l \frac{8}{7}$. The product of the two is $l \frac{9}{4}$. But the ratio for 3 is $l \frac{4}{3}$, canceling the remaining 4 in the denominator and one of the 3s in the numerator:
$l \displaystyle \frac{63}{32}\times\frac{8}{7}\times\frac{4}{3} = 3$
so $l 32\times 7\times 3 = 672$ is multiperfect with a ratio of 3.
Okay, let’s aim higher. Can we find a multiperfect number of order 5? Let’s start with a power of 2 that has a multiple of 5 for its sum of factors: $l \frac{4095}{2048}$. $l 4095=3^2\times 5\times 7\times 13$, so for our second ratio let’s use $l \frac{14}{13}$. That leaves us with $l \frac{3^2\times 5\times 7^2}{2^{10}}$.
Next if we use $l \frac{40}{27}$ (that’s $l 3^3$ in the denominator and $l 1+3+3^2+3^3$ in the numerator) that’ll cancel out three powers of 2, it’ll change the two powers of 3 in the numerator to one power in the denominator, and it’ll give us another 5 in the numerator:
$l \displaystyle \frac{4095}{2048}\times\frac{14}{13}\times\frac{40}{27} = \frac{5^2\times7^2}{2^7\times 3}$
Now we can cancel a couple more powers of 2 with $l \frac{20}{19}$:
$l \displaystyle \frac{4095}{2048}\times\frac{14}{13}\times\frac{40}{27}\times\frac{20}{19} = \frac{5^3\times7^2}{2^5\times 3\times 19}$
That 19 in the denominator looks unhealthy, but watch this: alakazam:
$l \displaystyle \frac{4095}{2048}\times\frac{14}{13}\times\frac{40}{27}\times\frac{20}{19}\times\frac{57}{49} = \frac{5^3}{2^5}$
And the crowd goes wild! We have two more powers of 5 than we need, though, so:
$l \displaystyle \frac{4095}{2048}\times\frac{14}{13}\times\frac{40}{27}\times\frac{20}{19}\times\frac{57}{49}\times\frac{31}{25} = \frac{5\times 31}{2^5}$
And then just one more step:
$l \displaystyle \frac{4095}{2048}\times\frac{14}{13}\times\frac{40}{27}\times\frac{20}{19}\times\frac{57}{49}\times\frac{31}{25}\times\frac{32}{31} = 5$
So there you are, in a few minutes with pencil and paper we get $l 2048\times 13\times 27\times 19\times 49\times 25\times 31$ is multiperfect with a ratio of 5. The hard part is multiplying those numbers together to get 518,666,803,200.$l ^*$
That isn’t the smallest order 5 multiperfect — you can find a smaller one starting with $l \frac{255}{128}$, and there’s another between those two. But I’ll let you chase those down.
* No, the really hard part is explicitly adding up the factors as a check:
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I may have let Python help with that.