Brady: Just a quick pre-video message; it would be very
helpful, although not absolutely essential, if you've watched our primes and primitive sets
video that we did earlier with Jared. I'll put links to it in all the usual places.
- Jared: We've come
up with a rule that assigns every set a number. - Almost like a fingerprint?
- Yeah exactly.
So today Brady we're going to be talking about uh the fingerprint numbers, we're going to give a nice
status update on some interesting conjectures. So if you're looking at uh primitive sets of odd
numbers then uh there is still a glimmer of hope that we can get a nice decomposition into odd
primitive sets.
- So Erdős was very interested in understanding how to maybe take collections of
numbers and compress them down into just one number or one value which can convey some added
information. And so in this case we're looking at the fingerprint numbers; so they're defined as
an infinite series, one fingerprint for each k, as the infinite series of 1 over n log n ranging
over numbers that have exactly k prime factors. So for example the first fingerprint number f1 is
the sum over all numbers that have exactly one prime factor - so in other words the prime numbers
themselves. So sum over primes p of one over p log p. So this is nothing more than just the sum of
1 + 2 log 2 + 1 over 3 log 3 + 1 over 5 log 5 and so on. So one term in the sum for each prime. This
series turns out to converge to about 1.636. So we're going to have a bunch of these series, these
fingerprint numbers, one for each k and there are many fascinating questions about these numbers -
some of which are due to Erdős, of which is the Erdős primitive set conjecture uh for some fans
on the channel. Another one which is- a could be viewed as a vast generalization is a conjecture
of two mathematicians Banks and Martin where they predicted so not only the first fingerprint number
the largest but we have this interesting chain of inequalities that f1 is bigger than f2 is bigger
than f3 is bigger than f4 and so on. This infinite chain of inequalities-
- (Brady: Two prime factors, all the numbers with three prime factors-)
- Exactly so each of these fingerprints corresponds to a whole collection of
numbers and the k fingerprint number looks at the numbers with exactly k prime factors. And so not
only in this conjecture uh we have this kind of beautiful relation, it actually applies in a much
more general setting. So I'm not going to maybe go in the full generality but another interesting
case is to consider odd numbers. So we can consider the same series, let's say call it gk, as the sum
of 1 over n log n ranging now over these odd numbers n - but as before they just have exactly k
prime factors. So this is another example of these kinds of sums in the conjecture; and the prediction
is that similarly these sums are decreasing at each step. So g1 is bigger than g2 is bigger than
g3 and so on. So so as I said before this is really kind of an important testing ground to see if we
have a good understanding and in the most general setting of the Banks and Martin conjecture really
says something important about the structure of primitive sets. So I want to maybe go to a new
piece of paper.
- (We don't know that they all get smaller but it looks-) That that is the- that
the expectation and so today as a status update we have some math news that in collaboration with
my co-authors Ofir Gorodetsky and Mo Dick Wong we are were to prove the odd version of the conjecture
holds eventually. So what do I mean by this, Brady? New result, I guess hot off the presses, this odd
version of the conjecture holds eventually. So let me maybe be a little bit more precise; so we show
that there's a large enough constant k, we show that gk satisfies the property. So gk is bigger
than gk + 1, is bigger than gk + 2 and so on forever. So we don't necessarily know that
the whole chain of inequalities is is working but we know at least far enough down the chain
that this starts to kick in.
- (All right, okay. Have-) (is there anywhere earlier in the chain where it doesn't work?)
- Uh no so we've also checked the- so this is a very good question. So we showed very
large k holding in this theorem; on the other hand we can just plug in on a computer and verify
the uh first cases. So this is a very uh pertinent question, so we can look at maybe a plot of what
these fingerprint numbers look like. So k up to 15 we can actually compute these directly on the
computer and we see for example g1 starts out a little bit larger than 0.9, g2 is a little bit
larger than 0.6, g3 is smaller, g4 is yet smaller, and at each step it seems to be getting smaller
and smaller. So what we actually prove in this result - so this is a essentially a corollary -we prove
we get an approximate formula for gk as being approximately 1/2. In the limit this uh sequence
of gks converges to 1/2; and this is a trend that we can already kind of eyeball in in the data but
we actually prove that it does so very fast at an exponential rate. So there's a secondary term which
decays as like 2 to the k. So you have this kind of nice exponential decay and the key point with this
result is that initially our series gk is defined as an infinite sum. So maybe from the beginning you
would think you would have to add up each term and you know wait a very long time, but this uh formula
here is giving us kind of essentially a shortcut, a way to kind of get to the answer at the at the end
of the line without having to do all the tedious uh computations. And then we can actually see uh
kind of conceptually why we're getting this uh uh data to really kind of bend towards 0.5.
- (You and your collaborators have proven that eventually) (this holds true; and you- and you've
just physically looked at the fact that it holds) (true at the start. At what point are you
confident that your proof kicks in? There's no set) (number your proof kicks in like like a million or?)
- Right so I'm sure- so so what I am confident about is that if we really worked very hard at ironing
out all these details we could maybe come up with some concrete number - maybe like a million or 10 to
the 100 or something like this. But definitely it would require some uh considerable effort to uh- so
so in the math lingo we say we want to make these results explicit. There's a a gap between looking
at very large numbers on the one hand and very small numbers which the the computer can handle
and we really want to make these results explicit in the sense that we want to fill in all the gaps
and have a complete understanding of the sequence. Uh Brady, so maybe uh you know the word explicit
nowadays takes on maybe an extra layer of meaning; so for example if you're listening to a song on
the radio you maybe hear your favorite song and it has some beeps in it you know that you know the
broadcasters had decided this wasn't ready for the clean version. When you hear these beeps that
really motivates me to, you know, go look the song up on my own like on YouTube and- to really hear
all the lyrics in the explicit version. And so in the same sense of it being explicit, as mathematicians
when we are presented with a kind of nice clean result like this that holds eventually, we're really kind
of motivated to want to go dig into the details and kind of iron everything out explicitly.
- (So you've shown me that the chain holds at the) (start and you've proven that the chain holds up
high. If you found one exception, one place) (where it doesn't hold, like you know at 1000
would that break your heart or would it excite you?) Okay so this is a- I guess maybe a a moment of
foreshadowing because we actually also have uh shown an analogous result for the the original
fingerprint numbers. This came as something of a surprise to us, we have an analogous result which
says that the the conjecture for the original fingerprint numbers fails eventually. So it the-
actually the opposite happens. So we prove that for k large the fk is less than fk + 1, is less than
fk+ 2 and so on forever more.
- (Where where does it switch?) Yeah so this is a a great uh phenomenon.
So so we have a kind of corresponding plot of the original fingerprint numbers; so we can see as we
saw f1 is about 1.6 a little larger f2 is a little bit larger than 1.1, and then at each step it seems
to go down and down but maybe eventually it starts to tail off towards 1. So maybe initially we
were thinking okay this could be uh the same kind of phenomenon we were seeing in the odd case.
But if we start to zoom in, if we really zoom into this area here, we start to see at around 5 and
6 it dips down to the minimum and then starts to rebound and actually go up. So what we actually
prove is is that for k large we get an approximate formula as before that fk is converging to 1. So
now if we include both the even numbers and the odd numbers these sums converge to 1; but they
do so from below, they go up. So now we have this minus sign where we also have a constant:
ck^2 over 2 to the k. So essentially we have this nice exponential decay to the limit but it's now coming
from the other side and this was a surprise to us. (And what happens now? Is this like a big
deal? Like-)
- So I I definitely would you know view this as being uh an important milestone in trying
to kind of nail down the full conjecture. So this this idea of making results explicit has a you
know very uh rich history in math; so maybe the best example uh that people are familiar with is
with the Goldbach conjecture. So uh for fans of the channel the Gold-the weak Goldbach conjecture
says that every odd number bigger than 5 can be written as a sum of three primes. And uh this
was you know famously made as a conjecture by Goldbach to Euler, which stumped Euler and and
for hundreds of years mathematicians tried to really prove this. And about 100 years ago uh
mathematician Vinogradov proved in 1937 that this conjecture of Goldbach holds eventually, holds
in the limit. And then it took over 80 years, until 2013, for Harald Helfgott to finally nail
down the solution completely. This is definitely a general arc, a story arc, that happens in in
mathematics in mathematics research that there is a milestone where one can hit to get a kind
of clean conceptual understanding uh but there's still a lot of work to do and new ideas to develop
sometimes to to really nail down the explicit version. ...99 consecutive numbers, none of which can
be prime numbers. And so you know that there has to be this gap between prime numbers of size at
least 100. But obviously you can replace 100 here with any number you like, and that shows that
there are arbitrary large gaps between prime numbers
