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A very unbelievable reality in regards to the quantity 37

A very unbelievable reality in regards to the quantity 37

2023-11-12 12:45:48

08 Nov 2023

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So I used to be on math stackexchange the opposite day, and I noticed a cute post
in search of a ebook which lists, for a lot of many integers, details that Ramanujan
may have advised Hardy if he’d taken a cab apart from 1729. A number of days in the past
OP answered their very own query, saying that the ebook in query was
Those Fascinating Numbers by Jean-Marie De Koninck. I made a decision to take
a look by means of it to see what sorts of details lie inside
(and likewise to see simply what number of integers are coated!). Not solely was I
overwhelmed by the variety of integers and the variety of details about them,
the preface already consists of one of many single wildest details I’ve ever heard,
and I’ve to speak about it right here! Right here’s a direct quote from the preface:

37, the median worth for the second prime issue of an integer;
thus the chance that the second prime issue of an integer
chosen at random is smaller than 37 is roughly $frac{1}{2}$;

My jaw was on the ground once I learn this, haha. First it sounded completely
unbelievable, since 37 is a tiny quantity within the grand scheme of issues. Then
it began to sound barely extra believable… In any case, about half of all
integers have $2$ as their smallest prime issue. It is smart that smaller
primes needs to be extra frequent among the many smallest components of numbers! However then
I assumed “how are you going to presumably show this!?”. I’m not a lot of an analytic
quantity theorist, however I do know that they’ve good estimates on loads of
details like this. I made a decision it might be enjoyable to attempt to discover and perceive a
proof of this reality, and likewise write some sage code to check it!

So then let’s go forward and do it ^_^

First, I feel, the sage code. I wish to know if this actually works!

“Obvoiusly” there’s no uniform distribution on the pure numbers, so
what does it even imply to decide on a “random” one? The best way the quantity theorists
often clear up this downside is by fixing a big quantity $N$ and
the chances if you choose a random quantity between $1$ and $N$. Then you definitely
have a look at the $N to infty$ restrict of those chances.

So for us, we’ll wish to first repair a big quantity $N$ after which work with
numbers $leq N$. For $N$ type of small, we are able to simply discover the second prime
issue of every quantity $leq N$ and examine the median!

After I first ran this code, it actually felt like magic, haha. What the
hell is occurring right here!?

The important thing thought, present in a paper of De Koninck and Tenenbaum,
is that we are able to compute the density of numbers whose second prime is $p$
(which the authors denote $lambda_2(p)$) by cleverly utilizing the concepts within the
Sieve of Eratosthenes!

Let’s do a easy instance to begin. What fraction of numbers have $5$ as
their second prime? Within the language of the paper, what’s $lambda_2(5)$?

Effectively it’s not arduous to see that the numbers whose second prime is $5$ are
these numbers whose prime factorization appears like

[2^a 3^0 5^b cdots]


[2^0 3^a 5^b cdots]

so we have to depend the density of numbers of those types.

However a quantity is of the primary kind ($2^a 3^0 5^b cdots$) if and provided that
it has an element of $2$, an element of $5$, and no components of $3$.

To carry this again to elementary faculty, we are able to spotlight all of
our numbers with an element of $2$

numbers with no components of $3$

and numbers with an element of $5$

Then the numbers whose prime factorization begins $2^a 3^0 5^b cdots$ are
precisely the numbers highlighted by all three of those colours!

It’s intuitively clear that $frac{1}{2}$ the numbers are blue,
$frac{2}{3}$ are orange, and $frac{1}{5}$ are pink.
So taken collectively,
$frac{1}{2} cdot frac{1}{5} cdot frac{2}{3} = frac{1}{15}$ of numbers
are of this way!

So now we’ve our fingers on the density of numbers of the shape $2^a 3^0 5^b$,
however this is just one of two ways in which $5$ might be the second smallest prime.
The same computation exhibits that
$left ( 1 – frac{1}{2} proper ) cdot frac{1}{3} cdot frac{1}{5} = frac{1}{30}$
of numbers are of the shape $2^0 3^a 5^b$.

It’s simple to see that these units are disjoint, so their densities add, and
$frac{1}{15} + frac{1}{30} = frac{1}{10}$ numbers have $5$ as their
second smallest issue!

Now with the warm-up out of the way in which, let’s see how we are able to compute
$lambda_2(p)$ for our favourite prime $p$!

We’ll play precisely the identical recreation. How can $p$ be the second smallest prime?
Precisely if the prime factorization appears like

[p^b q^a prod_{q neq r lt p} r^0]

for some $q lt p$.

However we are able to depend these densities as earlier than! For every selection of $q$, we all know
that $frac{1}{p}$ numbers are multiples of $p$, $frac{1}{q}$ are
multiples of $q$, and for every $r$ we all know $left (1 – frac{1}{r} proper )$
numbers are not multiples of $r$! For every $q$, then, we wish to land
within the intersection of all of those units, then we wish to sum over our
selections of $q$. Taken collectively, we see that

The density of numbers whose second prime is $p$ is

See Also

[lambda_2(p) =
sum_{q lt p}
prod_{q neq r lt p} left ( 1 – frac{1}{r} right )]

We will rearrange this to

lambda_2(p) =
left [ prod_{q lt p} left ( 1 – frac{1}{q} right ) right ] sum_{q lt p} frac{1}{q} left ( 1 – frac{1}{q} proper )^{-1})

As a cute train, write $lambda_k(p)$ for the density of numbers
whose $okay$th prime is $p$.

De Koninck and Tenenbaum point out in passing that

lambda_k(p) =
left [ prod_{k lt p} left ( 1 – frac{1}{q} right ) right ] s_{k-1}(p)]

the place
$s_j(p) = sum frac{1}{m}$
is a sum over all $m$ who’ve precisely $j$ prime components, all of that are $lt p$.

Are you able to show that this components is appropriate?

However keep in mind the purpose of all this! We wish to know the prime (p^*) in order that
half of all numbers have their second prime (leq p^*). That’s, in order that
the sum of densities

[lambda_2(2) + lambda_2(3) + lambda_2(5) + ldots + lambda_2(p^*)

However we are able to implement $lambda_2(-)$ and simply examine for which prime this

Once more we see that $37$ is the prime the place roughly half of all numbers
have one thing $leq 37$ as their first prime! So we’ve confirmed that
$37$ is the median second prime!

Additionally, this exhibits that we anticipate the precise density to be $approx .5002$.
If we set $N = 10^7$ within the code from the primary half
to get a greater approximation, we get $.5002501$, which is remarkably
near the reality!

As one other cute train – utilizing the concepts on this submit,
are you able to compute the median third prime?

As a (a lot) tougher train, are you able to get asymptotics for the way
the median $okay$th prime grows as a operate of $okay$?

Thanks for hanging out, all! This was a extremely enjoyable submit to put in writing up,
and I’m actually excited to share it with everyone!
This reality about $37$ was all I may take into consideration for like per week, haha.

I’ve extra weblog posts coming, after all, so I’ll see you all quickly!

Keep secure, and keep heat ^_^

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