so a bunch of math youtubers are doing this big crossover event where they talk about
their hashtag Mega Fav Numbers, that is, their favorite numbers over one million.
and, no, I wasn’t exactly “invited” to “participate”, but I’m also not really
a math youtuber so it makes sense.
I mean, yeah, this is my third consecutive video about math, but hey, this isn’t about
me, this is about numbers!
so, what’s my favorite number over a million? weird question, but why exactly is one million
the cutoff point? isn’t that a bit, I don’t know, decimal-centric? is there anything special
about the number one million that doesn’t rely on the way it’s represented in base
ten?
well, we can actually use the Online Encyclopedia of Integer Sequences to search for sequences
that contain any individual number, which is a good way of seeing what properties a
number has.
now, there are 1,703 sequences in the OEIS that contain the number one million.
so, the first result that shows up is that one million is a power of ten.
fair enough, but is there anything one million has going for it that a hundred thousand and
ten million don’t?
why this power of ten?
well, we can just exclude those from our search and find out.
well, now we’ve found that one million is a number to the power six, which means that
it’s both a perfect square and a perfect cube.
so, okay, that’s also pretty interesting, but again, why ten to the sixth and not nine
to the sixth or eleven to the sixth?
well, excluding those two, we can now see that one million is, the total number of subsets
of the n-th roots of 1 that add to zero?
what?
okay, so like, a “root of one” is a number where if you raise it to some integer exponent
you get one. and for real numbers, there’s only one and negative one where you can do
that.
but when you look at complex numbers, there’s a lot more.
so if you look at this sequence, right, you can see that when n=2, there’s two subsets
of the n-th roots of 1 that add to zero.
so what that means is that there’s two ways to write zero as the sum of some set of distinct
square roots of one.
so then what the fact that one million is integer number thirty six in this sequence
actually means is that there are one million ways to write zero as the sum of distinct
thirty sixth roots of one.
what was I talking about?
oh, right, the number one million.
so, my favorite mega number is 720,720.
the thing I like about it is that it’s a superior highly composite number.
I’ve spoken about SHCNs in the past, but I haven’t really talked about what they
actually are.
so let’s do that.
first, let’s talk about normal highly composite numbers, just because they’re easy to explain.
so, every positive integer has some amount of divisors, and some have more than others.
but which ones have the most divisors?
of course, there’s no upper bound on how many divisors a given integer can have, but
one easy way to formalize the idea that an integer has just a whole lot of divisors is
for it to have more divisors than any smaller positive integer.
if a positive integer has more divisors than any smaller positive integer, then it’s
a highly composite number.
and the highly composite numbers, you know, they’re pretty good! they’ve got a whole
bunch of divisors, but the superior highly composite numbers are way better.
so, for superior highly composite numbers, you’re not just counting divisors, you’re
doing something a bit more complicated.
so, you take your positive integer n, right, and you count how many divisors it has.
now, pick some small positive real number epsilon, and raise n to the power epsilon.
now, take the number of divisors of n and divide it by n to the epsilon.
if a number is a superior highly composite number, that means that there is some number
epsilon you can pick where doing that function to this number has an output that’s greater
than or equal to the output of that function for any other positive integer.
what that means is that there’s a way of counting divisors that takes into account
how small numbers tend to have fewer divisors than large numbers, and using this adjusted
way of counting factors, some numbers can be said to be “more composite” than all
other integers. and, in fact, there’s infinitely many of these superior highly composite numbers.
SHCNs have a lot of fun properties, but really it’s just great that they can be divided
in so many different ways, which brings us back to 720,720.
720,720 has 240 divisors.
it’s divisible by every integer from one through sixteen, every even integer from one
through thirty, and every threeven integer from one through twenty four.
oh, and hey, if you want a numbering system? base 720,720, which I call icosifetabakerheptelevenary,
for.
reasons, it would be absurdly difficult to learn, but it would also be extremely elegant!
there’s so many nice properties that would come with using it!
right, so here’s the thing.
720,720 is a great number.
but it’s also not bigger than a million.
while it is roughly the same size as a million, it is also very obviously less than a million,
and so I cannot call it my hashtag Mega Fav Number.
I mean, sure, I could just go up to the next superior highly composite number and talk
about 1,441,440, but like, it doesn’t even have that many more factors is the thing.
like, okay, the SHCN before 720,720 is 55,440, which has 120 divisors, and like, 720,720
has 240 divisors, right, so it’s literally twice as many, but then 1,441,440 only has
288? like ooh big deal that’s 48 more divisors from a number that’s twice the size.
like, come on.
but, sure, 720,720 is less than a million.
but that’s not the number in the thumbnail, is it? at what point will I bring up the number
23,240,400?
oh, I guess right now.
okay.
so, right, my hashtag Mega Fav Number is, in fact, two three two four oh four oh oh.
it is not, however, twenty three million two hundred forty thousand four hundred.
surprise! it’s base six! seven hundred twenty thousand seven hundred twenty in decimal is
equal to dozen three nif dozen four unexian four nif in seximal.
is this cheating?
yes.
do I care?
immensely.
but this just gets me back to that initial question.
why one million?
I suppose there just had to be some cutoff somewhere, and one million is a nice round
number.
but, is it?
I’d say that 2324 0400 is a much rounder number than 3323 3344. and, sure, 3323 3344
is quote unquote “bigger” than 2324 0400, but like, they’re the same order of magnitude.
I think it’s fine.
anyway, thanks for watching.
I guess if you’re into math stuff you should check out the Mega Fav Numbers playlist, and
if you’re coming here from the Mega Fav Numbers playlist and this is the first of
my videos you’re seeing then uh. thanks!
I guess.
I make a lot of stuff on this channel, including a handful of videos about math. uh, I made
a video about regular polyhedra pretty recently that did pretty well? you’d probably like
that if you like math. anyway, that’s it.
I don’t have a sign off.