Defining the Prime Gap function

[MevsEinstein]

TL;DR Summary

So I was creating a function $R(x)$ such that gives you the gap between the largest two primes less than or equal to $x$. I was trying to define it using the prime counting function, but I ran into problems

Hi PF!

I created a function $R(x)$ that gives the gap between the largest two primes less than or equal to $x$. To define it, I used this property: $$\pi(x+R(x))=\pi(x)+1$$ Which is true since the $x$ distance between $\pi(x)$ and $\pi(x)+1$ is $R(x)$. If we solve for $R(x)$ we get $$R(x) = \pi^{-1}(\pi(x)+1)-x$$ But $\pi^{-1}(x)$ isn't defined since $\pi(x)$ is a step function. Is there any other non-problematic way I can define $R(x)$?

 

[malawi_glenn]

Have you read "Introduction to analytic number theory" by Apostol?

 

[MevsEinstein]

Have you read "Introduction to analytic number theory" by Apostol?

No my library doesn't have it and my dad doesn't want me to buy books. Does it help?

 

[malawi_glenn]

Does it help?

The book? Yes it will teach you many things about analytic number theory. It is one of my fav math books.
Try these instead http://www2.math.uu.se/~astrombe/analtalt08/www_notes.pdf
https://faculty.math.illinois.edu/~hildebr/ant/main.pdf perhaps you could ask someone at your school to print these for you?

 

[MevsEinstein]

The book? Yes it will teach you many things about analytic number theory. It is one of my fav math books.
Try these instead http://www2.math.uu.se/~astrombe/analtalt08/www_notes.pdf
https://faculty.math.illinois.edu/~hildebr/ant/main.pdf perhaps you could ask someone at your school to print these for you?

Thank you very much!

 

[MevsEinstein]

I asked Wolfram to find the inverse function of $\frac{x}{\ln (x)}$ (which is an approximation for $\pi (x)$) and it gave me $-xW(-\frac{1}{x})$. So an approximation for $R(x)$ is $-(\pi(x)+1)W(-\frac{1}{\pi(x)+1}) - x$

 

[fresh_42]

...which is Lambert's W-function
https://en.wikipedia.org/wiki/Lambert_W_function
But it is only a function for $x>0.$

 

[MevsEinstein]

But it is only a function for $x>0.$

That's fine since we are only looking at positive integers.

 

[fresh_42]

That's fine since we are only looking at positive integers.

Yes, but you got $W(-1/x)$ which is a negative argument. If $x>0$ then $-1/x < 0.$

 

[MevsEinstein]

If $x>0$ then $-1/x < 0.$

OH. Well, the inverse prime function actually doesn't exist since $\pi(x)$ is a step function. So now what? Maybe if we think of $\pi^{-1}$ as a set of numbers and take the smallest one then we are fine?

 

[fresh_42]

OH. Well, the inverse prime function actually doesn't exist since $\pi(x)$ is a step function. So now what? Maybe if we think of $\pi^{-1}$ as a set of numbers and take the smallest one then we are fine?

Here is a graphic with approximations:
https://www.physicsforums.com/insights/the-history-and-importance-of-the-riemann-hypothesis/
The steps become tiny the greater the numbers are.

 

[malawi_glenn]

take the smallest one then we are fine?

Why don't you check for yourself and see if your R(x) gives desired result

 

[MevsEinstein]

Why don't you check for yourself

I don't know how to write the smallest value of $\pi^{-1}(x)$ in set notation. But I did go ahead and graph a few values of $R(x)$: https://www.desmos.com/calculator/vacrq5jxg1

 

[malawi_glenn]

$\pi^{-1}(x) = \min \left\{ y \in \mathbb{N} \, : \, \pi(y) = x \right\}$

 

[fresh_42]

Here is a book you could start with:
https://assets.openstax.org/oscms-prodcms/media/documents/Precalculus-OP_9wwF7YT.pdf

(Taken from https://openstax.org/subjects/math from Rice University.)

 

[MevsEinstein]

$\pi^{-1}(x) = \min \lbrace y \in \mathbb{N} \, : \, \pi(y) = x \rbrace$

Thanks! So $$R(x)= \min \lbrace y \in \mathbb{N} \, : \, \pi(y) = \pi(x) + 1 \rbrace - x$$

 

[MevsEinstein]

Here is a book you could start with:
https://assets.openstax.org/oscms-prodcms/media/documents/Precalculus-OP_9wwF7YT.pdf

(Taken from https://openstax.org/subjects/math from Rice University.)

This is why PF is amazing the people keep giving out resources. TYSM!

 

[malawi_glenn]

This is why PF is amazing the people keep giving out resources. TYSM!

I give my advanced and intersted high school students these links. The focus is on teaching how to read, construct and write proofs. Enjoy

“Mathematical reasoning”
https://scholarworks.gvsu.edu/cgi/viewcontent.cgi?article=1024&context=books
more info https://www.tedsundstrom.com/mathematical-reasoning-3

“Book of proof”
https://www.people.vcu.edu/~rhammack/BookOfProof/Main.pdf
more info https://www.people.vcu.edu/~rhammack/BookOfProof/

“A gentle introduction to the art of mathematics”
https://github.com/osj1961/giam/blob/master/GIAM.pdf?raw=true
more info https://osj1961.github.io/giam/

“An introduction to mathematical reasoning”
https://sites.math.washington.edu/~conroy/m300-general/ConroyTaggartIMR.pdf
more info https://sites.math.washington.edu/~conroy/2019/m300-win2019/index.htm

“Proofs and concepts - the fundamentals of abstract mathematics”
https://batch.libretexts.org/print/Finished/math-23870/Full.pdf
more info math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Proofs_and_Concepts

“Elementary Foundations: An Introduction to Topics in Discrete Mathematics”
https://batch.libretexts.org/print/Finished/math-83395/Full.pdf
more info https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Foundations

 

[fresh_42]

This is why PF is amazing the people keep giving out resources. TYSM!

Wait until you ask us for homework help. We've got this old-fashioned quirk to teach instead of providing ready-made solutions

 

[MevsEinstein]

$\pi^{-1}(x) = \min \lbrace y \in \mathbb{N} \, : \, \pi(y) = x \rbrace$

Thanks! So $$R(x)= \min{y \in \mathbb{N}$$

 

[MevsEinstein]

I give my advanced and intersted high school students these links.

I'm not even in high school yet

 

[MevsEinstein]

Enjoy

“Mathematical reasoning”
https://scholarworks.gvsu.edu/cgi/viewcontent.cgi?article=1024&context=books
more info https://www.tedsundstrom.com/mathematical-reasoning-3

“Book of proof”
https://www.people.vcu.edu/~rhammack/BookOfProof/Main.pdf
more info https://www.people.vcu.edu/~rhammack/BookOfProof/

“A gentle introduction to the art of mathematics”
https://github.com/osj1961/giam/blob/master/GIAM.pdf?raw=true
more info https://osj1961.github.io/giam/

“An introduction to mathematical reasoning”
https://sites.math.washington.edu/~conroy/m300-general/ConroyTaggartIMR.pdf
more info https://sites.math.washington.edu/~conroy/2019/m300-win2019/index.htm

“Proofs and concepts - the fundamentals of abstract mathematics”
https://batch.libretexts.org/print/Finished/math-23870/Full.pdf
more info math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Proofs_and_Concepts

“Elementary Foundations: An Introduction to Topics in Discrete Mathematics”
https://batch.libretexts.org/print/Finished/math-83395/Full.pdf
more info https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Foundations

I am getting dizzy

 

[malawi_glenn]

I'm not even in high school yet

Looks to me you think you are in graduate school :)

I am getting dizzy

hehe yeah, there is lot of stuff, but most of those follow the same format. Pick one and work it through, then pick another and see if there is anything new there. If you master the second pick, you can just browse through the other ones table of contents and see if there is anything there you have not encountered or mastered earlier.

Thanks! So $$R(x)= \min{y \in \mathbb{N}$$

no idea why you are typing this again

Thanks! So $$R(x)= \min \lbrace y \in \mathbb{N} \, : \, \pi(y) = \pi(x) + 1 \rbrace - x$$

 

[MevsEinstein]

I thought I didn't put the definition for R(x) so I typed it again on accident

 

[Office_Shredder]

If you're going to use the approximation $f(x)=x/ln(x)$, then you can just go all the way.

$f'(x)= \frac{\ln(x)-1}{\ln(x)^2}$
An interpretation of this is if you increase $x$ by $\epsilon$, then $f(x)$ increases by approximately $f'(x)\epsilon$. So to increase $f(x)$ by 1, you pick $\epsilon = 1/f'(x)$.

This can be an arbitrarily bad approximation, for example it will never tell you when there are twin primes.