A prime counting function I have never seen

[fresh_42]

Oh I am well aware of the Riemann hypothesis. It drives me nuts as I can’t deal with anything too abstract. Hence my problem with advanced mathematics. It doesn’t stop me from trying though.

Fermat claimed to have proven something and it took 350 years and a lot of very difficult mathematics to prove it. I don't say "again" since nobody truly believes Fermat had a proof. However, his claim led to the development of entire branches of mathematics.

Just make sure you won't try what already has been proven impossible:
A cube cannot be doubled by using a straightedge and compass.
A circle cannot be squared by using a straightedge and compass.
And an (arbitrary) angle cannot be split into three equal parts by using a straightedge and compass.

Here is a nice anecdote about the Riemann hypothesis:

It is not quite clear whether Hardy believed in God or was just superstitious. However, in any case, he believed God would do everything to make his life tough and complicated. One day he was on a journey back home to England. (I've heard it with Harald Bohr and Copenhagen, but also found Norway on the internet.) Anyway. He had to take a ship and the boat he got didn't look very trustworthy. Typically, he thought, why me?
So he sent a postcard before boarding - say to Bohr - claiming he had found the proof of Riemann's hypothesis.
When afterward asked why, he replied: Well, if the ship sank the proof would have been lost but I would have become the most famous mathematician of my generation. God won't allow this to happen. This way I only had to write Bohr another postcard in which I stated to have made a mistake.

 

[logistic_guy]

Thanks for this.

You're welcome

First off it was my mistake to say work as this is my hobby. Not career.

That's very interesting.

I was pretty upset that the golden ratio didn’t work out. But it did lead to the basis of this latest formula.

I call this golden ratio Magic. Last year in October a member threw a problem in this forum and we magically and precisely solved it by the golden ratio. Since you're interested in this golden ratio, you may want to take a look at here:

Brain teaser

Hi, I stumbled upon this question on Facebook and I tried to solve it unsuccessfully. I saw in the comments people talking about symmetry in their solutions but I didn't understand them. To me it seems there is insufficient info because the heights and widths of A and B share constraints, but...

www.freemathhelp.com

As you can probably tell from my vids I just play around and sometimes stumble across something interesting. Unfortunately it usually takes me awhile to figure out how I found it if I find it at all. But this formula does seem to be special. I’m just throwing it out there and hoping someone can do something with it.

If you want to progress further with your discovery and to be able to prove claims, I advise you to study number theory and abstract algebra (there are a lot about prime numbers there). This abstract algebra will sound confusing at the beginning, but once you proceed further, you'll find it fun to learn. Even if you find the first few chapters are easy to learn, don't skip a single page because the basics what make things become beautiful later.

I don't like to compliment people, but I will be honest with you, fresh_42 is a Genius professor. He has a great knowledge in mathematics, especially number theory and abstract algebra. If you decided to study them and had questions, I am sure that he would be so glad to answer your queries in a fantastic way. I myself have learnt a lot from him and still learning.

Also read and study every post and link he has given you carefully. Sometimes, you will need to spend a week or more reading to understand a specific idea from the book. The same idea can be understood in 15 minutes or less in a simple post written and explained by professor fresh_42.

There are also other great instructors on this forum whom I am sure will be glad to help you, but I’ve made professor fresh_42 my first priority. He is just a super Genius naturally.

Why the careless whisper reference? Did I have it playing in the background ground of one of my vids? lol

I think you created a playlist in your account and then forgot about it

Careless whisper is one of the first videos in that list.

 

[binbots]

Fermat claimed to have proven something and it took 350 years and a lot of very difficult mathematics to prove it. I don't say "again" since nobody truly believes Fermat had a proof. However, his claim led to the development of entire branches of mathematics.

Just make sure you won't try what already has been proven impossible:
A cube cannot be doubled by using a straightedge and compass.
A circle cannot be squared by using a straightedge and compass.
And an (arbitrary) angle cannot be split into three equal parts by using a straightedge and compass.

Here is a nice anecdote about the Riemann hypothesis:

It is not quite clear whether Hardy believed in God or was just superstitious. However, in any case, he believed God would do everything to make his life tough and complicated. One day he was on a journey back home to England. (I've heard it with Harald Bohr and Copenhagen, but also found Norway on the internet.) Anyway. He had to take a ship and the boat he got didn't look very trustworthy. Typically, he thought, why me?
So he sent a postcard before boarding - say to Bohr - claiming he had found the proof of Riemann's hypothesis.
When afterward asked why, he replied: Well, if the ship sank the proof would have been lost but I would have become the most famous mathematician of my generation. God won't allow this to happen. This way I only had to write Bohr another postcard in which I stated to have made a mistake.

lol. Sooo what you’re saying is I should have shown my equation with the caption: I solved the prime number theorem. Unfortunately I don’t have the time to write it all out.
Then I should have just disappeared. Ok next time I think of something that’s how I will play it. lol.

As for the Riemann hypothesis give me a day or two to respond. I have my own ideas on how that works. My ideas may not be correct but they are unique.

 

[binbots]

I forgot to mention that some theorems about the prime number density require the Riemann hypothesis to hold and I'm not certain whether Rosser's result does or does not. You can find a short description and some links about the relationship between [imath] \pi(x) [/imath] and the Riemann hypothesis here:

The History and Importance of the Riemann Hypothesis

The history of the Riemann hypothesis may be considered to start with the first mention of prime numbers in the Rhind Mathematical Papyrus around 1550 BC.

www.physicsforums.com

I made a video explaining why I believe this works. Hopefully you have 11 mins to spare.

 

[binbots]

You're welcome


That's very interesting.


I call this golden ratio Magic. Last year in October a member threw a problem in this forum and we magically and precisely solved it by the golden ratio. Since you're interested in this golden ratio, you may want to take a look at here:

Brain teaser

Hi, I stumbled upon this question on Facebook and I tried to solve it unsuccessfully. I saw in the comments people talking about symmetry in their solutions but I didn't understand them. To me it seems there is insufficient info because the heights and widths of A and B share constraints, but...

www.freemathhelp.com

If you want to progress further with your discovery and to be able to prove claims, I advise you to study number theory and abstract algebra (there are a lot about prime numbers there). This abstract algebra will sound confusing at the beginning, but once you proceed further, you'll find it fun to learn. Even if you find the first few chapters are easy to learn, don't skip a single page because the basics what make things become beautiful later.

I don't like to compliment people, but I will be honest with you, fresh_42 is a Genius professor. He has a great knowledge in mathematics, especially number theory and abstract algebra. If you decided to study them and had questions, I am sure that he would be so glad to answer your queries in a fantastic way. I myself have learnt a lot from him and still learning.

Also read and study every post and link he has given you carefully. Sometimes, you will need to spend a week or more reading to understand a specific idea from the book. The same idea can be understood in 15 minutes or less in a simple post written and explained by professor fresh_42.

There are also other great instructors on this forum whom I am sure will be glad to help you, but I’ve made professor fresh_42 my first priority. He is just a super Genius naturally.


I think you created a playlist in your account and then forgot about it

Careless whisper is one of the first videos in that list.

I was just wondering if you watched my newest video? It is getting a lot of attention. So much in fact that Im not even sure do it’s genuine or if I’m being trolled lol. I was hoping for your opinion.

 

[fresh_42]

I was just wondering if you watched my newest video? It is getting a lot of attention. So much in fact that Im not even sure do it’s genuine or if I’m being trolled lol. I was hoping for your opinion.

I still have the same opinion as in posts #8 and #15, but I will have a look. I have frozen my activities here due to members who have an unfriendly attitude I'm allergic to. I decided this website wasn't worth my time since some prefer to spread offenses and insults at me rather than arguments.

If I were looking for trolls I would have fun on FB with them. If there were someone who felt in charge of this site I would demand the deletion of my account and all my posts according to EU law.

 

[binbots]

I still have the same opinion as in posts #8 and #15, but I will have a look. I have frozen my activities here due to members who have an unfriendly attitude I'm allergic to. I decided this website wasn't worth my time since some prefer to spread offenses and insults at me rather than arguments.

If I were looking for trolls I would have fun on FB with them. If there were someone who felt in charge of this site I would demand the deletion of my account and all my posts according to EU law.

Yes I did notice a lot of friction around here. Not sure why considering there is an ignore option. It’s a shame because it’s a great place for people like me to come and get some expert input.
The video is getting a lot of attention. So hopefully something can come of it. Not sure how much more I can personally contribute as I am beyond my depth.

 

[simba087]

Dating for Sex.

Most realistic adult game

 

[fresh_42]

I was just wondering if you watched my newest video? It is getting a lot of attention. So much in fact that Im not even sure do it’s genuine or if I’m being trolled lol. I was hoping for your opinion.

I watched the video now and have two remarks. The first one is a technical one. Starting around 6:45 you introduce the formula

[math] \lim_{n \to \infty}\dfrac{\pi(e^{n+1})-\pi(e^n)}{\pi(e^n)-\pi(e^{n-1})}=e [/math]
I checked with the known formula [imath] \displaystyle{\lim_{n \to \infty}\pi(x)=\dfrac{x}{\log(x)}}. [/imath] I skipped the limits while calculating and put it in again at the end. So my lazy version reads [imath] \pi(e^n)=\dfrac{e^n}{n}. [/imath] Then I got

[math]\begin{array}{lll} \pi(e^n)-\pi(e^{n-1})&=\dfrac{e^n}{n}-\dfrac{e^{n-1}}{n-1}=\dfrac{(n-1)\cdot e \cdot e^{n-1}-n\cdot e^{n-1}}{n^2-n}=\dfrac{e^{n-1}}{n-1}\cdot \left(e-1-(e/n)\right)\\[12pt] \dfrac{\pi(e^n)-\pi(e^{n-1})}{\pi(e^{n+1})-\pi(e^{n})}&= \dfrac{e^{n-1}}{n-1}\dfrac{n}{e^n}\cdot \dfrac{e-1-(e/n)}{e-1-(e/(n+1))}\stackrel{\lim_{n \to \infty}}{\longrightarrow } \displaystyle{\lim_{n \to \infty}\dfrac{1}{(1-(1/n))e}}=\dfrac{1}{e} \end{array}[/math]
So, yes, your observation is correct. This leads me to my second remark. The German Wikipedia has this nice picture

which I think doesn't need a translation. Replace 'ie' by 'y', 'k' and 'z' by 'c' and 'allgemein' by 'general' and you have it in English. Your reasoning is on the inductive side which is a kind of physical approach whereas results in mathematics are on the deductive side. Physics cannot answer the question 'why'. Here is an interesting interview about the 'why' question in physics with Richard Feynman:

Mathematics is all about answering 'why'. It all starts with a collection of axioms, e.g. for set theory and arithmetic, a kind of god-given set of facts that are commonly accepted to be true, e.g. "Every natural number has a successor." You don't have to agree with them. But if not, you will need another collection of axioms and you get a different kind of mathematics. Every theorem in mathematics is finally a chain of deductions based on these assumed facts called axioms.

That's the main problem with observations. It is impossible to get from the inductive side onto the deductive side by empirical methods. An interesting example of this is the Riemann hypothesis. It has been tested for 10,000,000,000,000 many numbers but we still consider it as unproven.

Yes I did notice a lot of friction around here. Not sure why considering there is an ignore option. It’s a shame because it’s a great place for people like me to come and get some expert input.

I thought this would be a place with less restrictions, not more. I came here from another website that strictly forbids any "personal theories" which is fine if you want to study something, but problematic if you want to have a comment about an idea. They would rather ban people quickly than give them a bit more leash. What I found, however, was a witch hunt on @logistic_guy and insults if you dare to defend him. Not the kind of tolerance I have been looking for.

 

[binbots]

I watched the video now and have two remarks. The first one is a technical one. Starting around 6:45 you introduce the formula

[math] \lim_{n \to \infty}\dfrac{\pi(e^{n+1})-\pi(e^n)}{\pi(e^n)-\pi(e^{n-1})}=e [/math]
I checked with the known formula [imath] \displaystyle{\lim_{n \to \infty}\pi(x)=\dfrac{x}{\log(x)}}. [/imath] I skipped the limits while calculating and put it in again at the end. So my lazy version reads [imath] \pi(e^n)=\dfrac{e^n}{n}. [/imath] Then I got

[math]\begin{array}{lll} \pi(e^n)-\pi(e^{n-1})&=\dfrac{e^n}{n}-\dfrac{e^{n-1}}{n-1}=\dfrac{(n-1)\cdot e \cdot e^{n-1}-n\cdot e^{n-1}}{n^2-n}=\dfrac{e^{n-1}}{n-1}\cdot \left(e-1-(e/n)\right)\\[12pt] \dfrac{\pi(e^n)-\pi(e^{n-1})}{\pi(e^{n+1})-\pi(e^{n})}&= \dfrac{e^{n-1}}{n-1}\dfrac{n}{e^n}\cdot \dfrac{e-1-(e/n)}{e-1-(e/(n+1))}\stackrel{\lim_{n \to \infty}}{\longrightarrow } \displaystyle{\lim_{n \to \infty}\dfrac{1}{(1-(1/n))e}}=\dfrac{1}{e} \end{array}[/math]
So, yes, your observation is correct. This leads me to my second remark. The German Wikipedia has this nice picture

which I think doesn't need a translation. Replace 'ie' by 'y', 'k' and 'z' by 'c' and 'allgemein' by 'general' and you have it in English. Your reasoning is on the inductive side which is a kind of physical approach whereas results in mathematics are on the deductive side. Physics cannot answer the question 'why'. Here is an interesting interview about the 'why' question in physics with Richard Feynman:

Mathematics is all about answering 'why'. It all starts with a collection of axioms, e.g. for set theory and arithmetic, a kind of god-given set of facts that are commonly accepted to be true, e.g. "Every natural number has a successor." You don't have to agree with them. But if not, you will need another collection of axioms and you get a different kind of mathematics. Every theorem in mathematics is finally a chain of deductions based on these assumed facts called axioms.

That's the main problem with observations. It is impossible to get from the inductive side onto the deductive side by empirical methods. An interesting example of this is the Riemann hypothesis. It has been tested for 10,000,000,000,000 many numbers but we still consider it as unproven.



I thought this would be a place with less restrictions, not more. I came here from another website that strictly forbids any "personal theories" which is fine if you want to study something, but problematic if you want to have a comment about an idea. They would rather ban people quickly than give them a bit more leash. What I found, however, was a witch hunt on @logistic_guy and insults if you dare to defend him. Not the kind of tolerance I have been looking for.

Thank you for this in-depth response. I will need sometime to process it…..

 

[binbots]

I watched the video now and have two remarks. The first one is a technical one. Starting around 6:45 you introduce the formula

[math] \lim_{n \to \infty}\dfrac{\pi(e^{n+1})-\pi(e^n)}{\pi(e^n)-\pi(e^{n-1})}=e [/math]
I checked with the known formula [imath] \displaystyle{\lim_{n \to \infty}\pi(x)=\dfrac{x}{\log(x)}}. [/imath] I skipped the limits while calculating and put it in again at the end. So my lazy version reads [imath] \pi(e^n)=\dfrac{e^n}{n}. [/imath] Then I got

[math]\begin{array}{lll} \pi(e^n)-\pi(e^{n-1})&=\dfrac{e^n}{n}-\dfrac{e^{n-1}}{n-1}=\dfrac{(n-1)\cdot e \cdot e^{n-1}-n\cdot e^{n-1}}{n^2-n}=\dfrac{e^{n-1}}{n-1}\cdot \left(e-1-(e/n)\right)\\[12pt] \dfrac{\pi(e^n)-\pi(e^{n-1})}{\pi(e^{n+1})-\pi(e^{n})}&= \dfrac{e^{n-1}}{n-1}\dfrac{n}{e^n}\cdot \dfrac{e-1-(e/n)}{e-1-(e/(n+1))}\stackrel{\lim_{n \to \infty}}{\longrightarrow } \displaystyle{\lim_{n \to \infty}\dfrac{1}{(1-(1/n))e}}=\dfrac{1}{e} \end{array}[/math]
So, yes, your observation is correct. This leads me to my second remark. The German Wikipedia has this nice picture

which I think doesn't need a translation. Replace 'ie' by 'y', 'k' and 'z' by 'c' and 'allgemein' by 'general' and you have it in English. Your reasoning is on the inductive side which is a kind of physical approach whereas results in mathematics are on the deductive side. Physics cannot answer the question 'why'. Here is an interesting interview about the 'why' question in physics with Richard Feynman:

Mathematics is all about answering 'why'. It all starts with a collection of axioms, e.g. for set theory and arithmetic, a kind of god-given set of facts that are commonly accepted to be true, e.g. "Every natural number has a successor." You don't have to agree with them. But if not, you will need another collection of axioms and you get a different kind of mathematics. Every theorem in mathematics is finally a chain of deductions based on these assumed facts called axioms.

That's the main problem with observations. It is impossible to get from the inductive side onto the deductive side by empirical methods. An interesting example of this is the Riemann hypothesis. It has been tested for 10,000,000,000,000 many numbers but we still consider it as unproven.



I thought this would be a place with less restrictions, not more. I came here from another website that strictly forbids any "personal theories" which is fine if you want to study something, but problematic if you want to have a comment about an idea. They would rather ban people quickly than give them a bit more leash. What I found, however, was a witch hunt on @logistic_guy and insults if you dare to defend him. Not the kind of tolerance I have been looking for.

Well first off I would like to say thank for all the information you have put together for me. I will probably make a follow up video and use your maths in it. Obviously with your consent. It does add a lot of validity to this idea and hopefully it will lead to more people with different skills sets investigation it.
As for my methods being inductive, I am well aware of that. In fact my whole logic behind my method is that I could probe ancient mathematics that many just over look now, but do it with a calculator. To maybe come across something we might have over looked. Like giving Euler an iPhone and seeing what he kind find probing mathematical realms like e^1000000. So yes the method may not be ideal but I don’t think it’s still not worth doing. I mean the Riemann hypothesis is still only an hypothesis but yet it has done wonders in just trying to be proven.
Thank you for the Feynman video. It has been awhile since I have watched his interviews and lectures. I am a big fan. The only thing he preaches that I don’t prescribe to is his shut up and calculate idea. For advanced and applied math for sure this is applicable. But for me while all these math wizards do all these amazing things with these mathematical tools I’m too busy staring at the tool itself wondering what it is made of.
As for axioms I believe my axioms are the properties of bounded and unbounded infinities. Obviously not your typical axioms but I’m not sure how they can be disputed. I mean if we are going to keep expanding our maths we will have to explore new axioms to see where they lead us.
As for proving my idea I’m leaning more to the side that that it is unprovable. Or even another level deeper that it’s impossible to even know if it’s provable or unprovable. This all being a consequence of my axioms.
As for the equation itself. It may not be possible to prove it. But it still might be the best arithmetic approximation we maybe able to come up with. The fact that the bounded and unbounded infinities begin with e and ultimately end in a ratio of prime density of e in the same amount of steps seems too much of a coincidence not to be accurate. Hence my desire to get as many smart people to investigate it further.

I can’t believe another site wouldn’t allow personal ideas. That seems very odd. Especially with math when if an idea is false it’s usually stomped out pretty quickly. It wouldn’t be like a false opinion that would permeate the community for years on end. But if an idea is correct could give a lot of truthful insight.
Yes the drama here seems pretty unnecessary. Logistic_guy may not be your traditional math guy (obviously neither am I) but he’s harmless. Plus this isn’t Harvard. This is a forum on the internet. Where all kinds of people and ideas can come together without formal discourse.
Anyways I hope you stick around cause you have been a great help to me so far.

 

[fresh_42]

Well first off I would like to say thank for all the information you have put together for me. I will probably make a follow up video and use your maths in it. Obviously with your consent. It does add a lot of validity to this idea and hopefully it will lead to more people with different skills sets investigation it.
As for my methods being inductive, I am well aware of that. In fact my whole logic behind my method is that I could probe ancient mathematics that many just over look now, but do it with a calculator. To maybe come across something we might have over looked. Like giving Euler an iPhone and seeing what he kind find probing mathematical realms like e^1000000. So yes the method may not be ideal but I don’t think it’s still not worth doing. I mean the Riemann hypothesis is still only an hypothesis but yet it has done wonders in just trying to be proven.

Fermat's last theorem established an entire branch of mathematics, the algebraic part of number theory. Primes have fascinated us humans for a long. They have a lot of strange properties. I mean

[math] \dfrac{\pi^2}{6} =\displaystyle{\prod_{p\text{ prime} } \dfrac{p^2}{(p-1)(p+1)}} [/math]
is so strange (to me). I can read and understand the proof but cannot grasp its mystery.

Thank you for the Feynman video. It has been awhile since I have watched his interviews and lectures. I am a big fan. The only thing he preaches that I don’t prescribe to is his shut up and calculate idea. For advanced and applied math for sure this is applicable. But for me while all these math wizards do all these amazing things with these mathematical tools I’m too busy staring at the tool itself wondering what it is made of.
As for axioms I believe my axioms are the properties of bounded and unbounded infinities. Obviously not your typical axioms but I’m not sure how they can be disputed. I mean if we are going to keep expanding our maths we will have to explore new axioms to see where they lead us.

You can do strange things with infinities so we must be very careful with them. Many people treat them like they treat numbers and write things like [imath] 1/00\infty [/imath] which is not only wrong, it also leads to further misconceptions. Hilbert's hotel is a nice example of what you can do with infinities, or the fact that you can paint out a square with a line: something two-dimensional colored with something one-dimensional.

Axioms are difficult, too. Hilbert's second problem simply asked whether the axioms of arithmetic are without contradictions. And the answer is, that this cannot be decided within arithmetic. Even such a seemingly simple question leads us deep into logical conflicts.

As for proving my idea I’m leaning more to the side that that it is unprovable. Or even another level deeper that it’s impossible to even know if it’s provable or unprovable. This all being a consequence of my axioms.
As for the equation itself. It may not be possible to prove it. But it still might be the best arithmetic approximation we maybe able to come up with. The fact that the bounded and unbounded infinities begin with e and ultimately end in a ratio of prime density of e in the same amount of steps seems too much of a coincidence not to be accurate. Hence my desire to get as many smart people to investigate it further.

I think that your formula is equivalent to the known one [imath] \pi(x) \sim x/log(x), [/imath] i.e. that the two differ only by a sequence that tends to zero. That would lead to considerations about how much these differ from the true value of [imath] pi(x) [/imath] for which we do not have a closed formula and for which values of [imath] x. [/imath]. All I could find on this subject were statements like

[math] \dfrac{x}{\log(x) -1/2}<\pi(x)< \dfrac{x}{\log(x) -3/2} [/math]
or similar.

I can’t believe another site wouldn’t allow personal ideas. That seems very odd. Especially with math when if an idea is false it’s usually stomped out pretty quickly
It wouldn’t be like a false opinion that would permeate the community for years on end. But if an idea is correct could give a lot of truthful insight.

The rule is meant to avoid useless discussions about things that have already been proven impossible like squaring a circle, or people claiming that they can prove the Riemann hypothesis. The odds of finding such a proof on the Internet are factually zero and any dealing with it is wasting everybody's time.

It is a physics website, and here is where the discrepancy between the inductive logic of physics and the deductive logic of mathematics creates the problem. They don't want to discuss ether, time travel or flat earth. Statements in mathematics are clearer: unknown, proven false, proven correct, or as in my example of arithmetic, undecidable. It is easier in mathematics to take a personal theory to the test. On the other hand, there are flat earthers and circle-squarers out there despite known results and you surely don't want to debate with those. However, the rule against those "theories" leads to side effects since it is people who decide what is a personal theory and what is just an idea. One, but not the only side effect is to ban people who phrased their ideas as statements instead of questions. It is the distinction between false claims and curiosity. But who thinks that far when he wants to discuss their idea?

Yes the drama here seems pretty unnecessary. Logistic_guy may not be your traditional math guy (obviously neither am I) but he’s harmless. Plus this isn’t Harvard. This is a forum on the internet. Where all kinds of people and ideas can come together without formal discourse.

I couldn't agree more.

 

[binbots]

Fermat's

Fermat's last theorem established an entire branch of mathematics, the algebraic part of number theory. Primes have fascinated us humans for a long. They have a lot of strange properties. I mean

π26=∏p primep2(p−1)(p+1) \dfrac{\pi^2}{6} =\displaystyle{\prod_{p\text{ prime} } \dfrac{p^2}{(p-1)(p+1)}} 6π2=p prime∏(p−1)(p+1)p2
is so strange (to me). I can read and understand the proof but cannot grasp its mystery.

Agreed. The mystery of the primes has fascinated me for over 20 years now. The fact that it’s all about combining patterns yet they are random has kept me hooked all this time. Until I realized that primes are what’s left over after you combine an infinite amount of patterns therefore would not have a pattern themselves. Which I have never heard of them described that way but it makes sense to me.
last theorem established an entire branch of mathematics, the algebraic part of number theory. Primes have fascinated us humans for a long. They have a lot of strange properties. I mean

You can do strange things with infinities so we must be very careful with them. Many people treat them like they treat numbers and write things like 1/00∞ 1/00\infty 1/00∞ which is not only wrong, it also leads to further misconceptions. Hilbert's hotel is a nice example of what you can do with infinities, or the fact that you can paint out a square with a line: something two-dimensional colored with something one-dimensional.

Axioms are difficult, too. Hilbert's second problem simply asked whether the axioms of arithmetic are without contradictions. And the answer is, that this cannot be decided within arithmetic. Even such a seemingly simple question leads us deep into logical conflicts.

i guess any axioms pertaining to infinities is a big stretch. But still stuff I love to ponder about

I think that your formula is equivalent to the known one π(x)∼x/log(x), \pi(x) \sim x/log(x), π(x)∼x/log(x), i.e. that the two differ only by a sequence that tends to zero. That would lead to considerations about how much these differ from the true value of pi(x) pi(x) pi(x) for which we do not have a closed formula and for which values of x. x. x.. All I could find on this subject were statements like

xlog⁡(x)−1/2<π(x)<xlog⁡(x)−3/2 \dfrac{x}{\log(x) -1/2}<\pi(x)< \dfrac{x}{\log(x) -3/2} log(x)−1/2x<π(x)<log(x)−3/2x
or similar.

It seems if my equation can do anything it can increase the lower bound. I just don’t know where to begin in proving that.

It is a physics website, and here is where the discrepancy between the inductive logic of physics and the deductive logic of mathematics creates the problem. They don't want to discuss ether, time travel or flat earth. Statements in mathematics are clearer: unknown, proven false, proven correct, or as in my example of arithmetic, undecidable. It is easier in mathematics to take a personal theory to the test. On the other hand, there are flat earthers and circle-squarers out there despite known results and you surely don't want to debate with those. However, the rule against those "theories" leads to side effects since it is people who decide what is a personal theory and what is just an idea. One, but not the only side effect is to ban people who phrased their ideas as statements instead of questions. It is the distinction between false claims and curiosity. But who thinks that far when he wants to discuss their idea?

I actually have a physics idea that gets a lot of positive feedback. But then someone got upset with me cause I started it as if it were a fact. I told him that I’m sharing an idea on how to combine QM and GR in a YouTube comment. I didn’t think I had to prefix it with I believe at the beginning cause I thought it was obvious. Lol

I watched the video now and have two remarks. The first one is a technical one. Starting around 6:45 you introduce the formula

lim⁡n→∞π(en+1)−π(en)π(en)−π(en−1)=e \lim_{n \to \infty}\dfrac{\pi(e^{n+1})-\pi(e^n)}{\pi(e^n)-\pi(e^{n-1})}=e n→∞limπ(en)−π(en−1)π(en+1)−π(en)=

I have shifted my focus more to this now since you have proven that it is correct. Once again thank you. I think I found out how multiply the lower section to get the next.

Is there anything I can do with this? As in simplify it or even turn it into a function?

. But not too much because it inevitably will drive you insane. It always lead me the realization that math is just counting an infinite amount of zeros.

[math] \dfrac{\pi^2}{6} =\displaystyle{\prod_{p\text{ prime} } \dfrac{p^2}{(p-1)(p+1)}} [/math]
is so strange (to me). I can read and understand the proof but cannot grasp its mystery.





I think that your formula is equivalent to the known one [imath] \pi(x) \sim x/log(x), [/imath] i.e. that the two differ only by a sequence that tends to zero. That would lead to considerations about how much these differ from the true value of [imath] pi(x) [/imath] for which we do not have a closed formula and for which values of [imath] x. [/imath]. All I could find on this subject were statements like

[math] \dfrac{x}{\log(x) -1/2}<\pi(x)< \dfrac{x}{\log(x) -3/2} [/math]
or similar.



The rule is meant to avoid useless discussions about things that have already been proven impossible like squaring a circle, or people claiming that they can prove the Riemann hypothesis. The odds of finding such a proof on the Internet are factually zero and any dealing with it is wasting everybody's time.

It is a physics website, and here is where the discrepancy between the inductive logic of physics and the deductive logic of mathematics creates the problem. They don't want to discuss ether, time travel or flat earth. Statements in mathematics are clearer: unknown, proven false, proven correct, or as in my example of arithmetic, undecidable. It is easier in mathematics to take a personal theory to the test. On the other hand, there are flat earthers and circle-squarers out there despite known results and you surely don't want to debate with those. However, the rule against those "theories" leads to side effects since it is people who decide what is a personal theory and what is just an idea. One, but not the only side effect is to ban people who phrased their ideas as statements instead of questions. It is the distinction between false claims and curiosity. But who thinks that far when he wants to discuss their idea?



I couldn't agree more.

 

[fresh_42]

View attachment 39160
Is there anything I can do with this? As in simplify it or even turn it into a function?

I don't really have an idea. Let us write [imath] e^x=a [/imath] for a moment. Then we have
[math] a-\pi\left(\dfrac{a}{e}\right)\dfrac{e}{e^{1/x}}=\pi(a\cdot e)-\pi(a) .[/math]This means we have the value of an unknown function [imath] \pi [/imath] at three very different values, namely [imath] a/e\, , \,a\cdot e\, , \,a [/imath] plus a nasty factor for which we only know that it tends to [imath] e [/imath] for large values of [imath] x. [/imath] I thought of differentiating it to see whether I could see something but that factor creates a mess. Since all locations differ by a factor [imath] e, [/imath] the natural question would be if we can make any assumptions about the behavior on products, i.e. what is [imath] \pi(x\cdot y) [/imath] in terms of [imath] \pi(x) [/imath] and [imath] \pi(y). [/imath] E.g. exponentiation has the functional equations [imath] a^{x+y}=a^x\cdot a^y [/imath] and [imath] a^{x\cdot y}=\left(a^x\right)^y. [/imath] In order to derive something, we need to have something and be it an assumption about the behavior of [imath] \pi. [/imath]


The only thing which comes to mind is a historical remark. Mathematicians in the 17th century, like Fermat, usually were jurists, theologists, or people of another profession in real life. And they had to write letters to share their ideas with colleagues. Some of them, and if I remember correctly, Fermat was one of them, wrote letters in which they shared a problem they already had a solution for only to figure out whether their opponent could find it. But, of course, there also had been normal correspondence. It wasn't always nasty. Our discussion here reminds me a bit of those letters.

 

[fresh_42]

I have looked at it for small values and have set [imath] \pi(1)=0\, , \,\pi(e)=1. [/imath] Then
[math]\begin{array}{lll} \pi(e^{x+1})&=e^x+\pi(e^x)-\pi\left(e^{x-1}\right)\cdot e^{1-1/x}\\ \pi(e)&=1\\ \pi(e^2)&=e+1\\ \pi(e^3)&=e^2+e+1-\sqrt{e}\\ \pi(e^4)&=e^3+e^2+e+1-\sqrt{e}-(e+1)\cdot \sqrt[3]{e^2}=e^3+e^2+e+1-\sqrt{e}-\sqrt[3]{e^2}-\sqrt[3]{e^5} \end{array}[/math]This shows the difficulty. The correction term [imath] e^{1-1/x} [/imath] makes it more complicated with every step.

 

[binbots]

The only thing which comes to mind is a historical remark. Mathematicians in the 17th century, like Fermat, usually were jurists, theologists, or people of another profession in real life. And they had to write letters to share their ideas with colleagues. Some of them, and if I remember correctly, Fermat was one of them, wrote letters in which they shared a problem they already had a solution for only to figure out whether their opponent could find it. But, of course, there also had been normal correspondence. It wasn't always nasty. Our discussion here reminds me a bit of those letters.

Exactly. This is how I pictured forums would be like. All kinds of people with different experiences and therefore different ideas coming together without judgement to brainstorm ideas no matter how outlandish they may be. It’s hard to come up with original concepts when you’re not willing to entertain anything outside the norm. But I guess even the math world like most things these days has become very jaded by the social media era. I always thought this kind of easy communication would have been amazing but it has actually become the opposite.

Thanks again for all your insight. I can see what a mess it is becoming. Let me attempt to explain to you what I’m visualizing and trying to construct.

As you can see my idea kind of works. Unfortunately it does lack a bit behind. But maybe not indefinitely.

Even if it’s not correct im just showing what I’m trying to accomplish. For example….
Now this example doesn’t work at all. But there has to be a way to combine them correctly. It’s also made harder by the fact we are not dealing with pi(x) but rather pi(e^x)-pi(e^(x-1). But that does seem surmountable.
I also think we can break up the 1/xsqrt(e))•e more by diving up by (e^x)-e^(x-1). So we can see how many steps it took to get to the next section.
Sorry if this is confusing or doesn’t make sense. But when you’re teaching yourself and probing things that you have never seen before or may have never been probed by anyone before it can get confusing. There just must be away….

 

[fresh_42]

That's my suspicion, that your formula is basically equivalent to other formulas in the sense that the difference vanishes the larger [imath] x [/imath] gets, but I haven't proven it formally. I think that the proof of the prime number theorem [imath] \pi(x) \approx \dfrac{x}{\log(x)} [/imath] is not too complicated, although it looks as if the first proofs used the Riemannian zeta function which isn't so easy. But my French is more than rusty and the language in papers from 1896 is all but modern. In the case you're formula is really equivalent to the prime number theorem, then the question arises whether it has stronger or weaker approximation properties than the known approximations.

I have found an interesting quotation about a later proof of the prime number theorem:

”The resulting proof of the prime number theorem is short, beautiful, and understandable without any knowledge of number theory or complex function theory beyond Cauchy’s theorem; it should be known to every mathematician.“​

Correction: It looks as if it is easy for mathematicians! Here is the graphic summary of the proof



I will refrain from translating here because the message is clear as it is: Do not trust mathematicians when they say that something is "understandable".

The version (of the proof of the prime number theorem) where I took this image from is only one page long, but the preparation is seventeen pages long! And it takes a bit of function theory.

Give me some time to consider your formula [imath]\dfrac{ \pi(e^n)}{e^{n-1}}=\prod_{k=2}^n \dfrac{1}{\sqrt[k]{e}} .[/imath]

Doesn't look wrong at first glance, although the convergence is a bit slow:

(1/20) - e* prod (1/(e^(1/k))) from k=2 to 20 - Wolfram|Alpha

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

www.wolframalpha.com

(1/100) - e* prod (1/(e^(1/k))) from k=2 to 100 - Wolfram|Alpha

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

www.wolframalpha.com

You get faster if you replace [imath] e [/imath] by a larger number, e.g. [imath] 10. [/imath]

(1/20) - 10* prod (1/(10^(1/k))) from k=2 to 20 - Wolfram|Alpha

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

www.wolframalpha.com

 

[fresh_42]

The formula [imath] \pi(e^x)=\dfrac{e}{\sqrt[2]{e}}\cdot \dfrac{e}{\sqrt[3]{e}}\cdots\dfrac{e}{\sqrt[x]{e}}[/imath] means [imath] \dfrac{\pi(e^x)}{e^x}=e\cdot \dfrac{1}{\sqrt[2]{e}}\cdot \dfrac{1}{\sqrt[3]{e}}\cdots\dfrac{1}{\sqrt[x]{e}}.[/imath] If we assume that [imath] \pi(x)\sim\dfrac{x}{\log x} [/imath] then [imath] \dfrac{\pi(e^x)}{e^x}\sim \dfrac{1}{\log e^x}=\dfrac{1}{x} [/imath] and we have to compare the product of roots with the value [imath] 1/x. [/imath] Now the series expansion at infinity for the product of roots is

[math]\begin{array}{lll} \displaystyle{e\cdot \prod_{k=2}^n \dfrac{1}{\sqrt[k]{e}}=\dfrac{e^{2-\gamma}}{n} - \dfrac{e^{2-\gamma}}{2n^2} + \dfrac{5e^{2-\gamma}}{24n^3} -\dfrac{e^{2-\gamma}}{16n^4} + \dfrac{47e^{2-\gamma}}{5760n^5}+ O\left(\dfrac{1}{n^6}\right)\approx \dfrac{e^{1.423}}{n}+ O\left(n^{-2}\right)\approx\dfrac{ 4.148655621}{n}+O\left(n^{-2}\right)} \end{array}[/math]
This means that you have a systematic error of a bit over [imath] \dfrac{3}{x} [/imath] in your formula. This becomes smaller with larger values of [imath] x. [/imath] Nevertheless it is an error. [imath] \gamma \approx 0.57721\ldots[/imath] is the Euler-Mascheroni constant, defined as [math] \displaystyle{\gamma= \lim_{n \to \infty} \left(-\log n +\sum_{k=1}^n\dfrac{1}{k}\right)}.[/math]
You should divide your formula for [imath] \pi(x) [/imath] by [imath] e^{2-\gamma} [/imath] to get an error of the magnitude [imath] 1/x^2. [/imath]

 

[binbots]

The formula [imath] \pi(e^x)=\dfrac{e}{\sqrt[2]{e}}\cdot \dfrac{e}{\sqrt[3]{e}}\cdots\dfrac{e}{\sqrt[x]{e}}[/imath] means [imath] \dfrac{\pi(e^x)}{e^x}=e\cdot \dfrac{1}{\sqrt[2]{e}}\cdot \dfrac{1}{\sqrt[3]{e}}\cdots\dfrac{1}{\sqrt[x]{e}}.[/imath] If we assume that [imath] \pi(x)\sim\dfrac{x}{\log x} [/imath] then [imath] \dfrac{\pi(e^x)}{e^x}\sim \dfrac{1}{\log e^x}=\dfrac{1}{x} [/imath] and we have to compare the product of roots with the value [imath] 1/x. [/imath] Now the series expansion at infinity for the product of roots is

[math]\begin{array}{lll} \displaystyle{e\cdot \prod_{k=2}^n \dfrac{1}{\sqrt[k]{e}}=\dfrac{e^{2-\gamma}}{n} - \dfrac{e^{2-\gamma}}{2n^2} + \dfrac{5e^{2-\gamma}}{24n^3} -\dfrac{e^{2-\gamma}}{16n^4} + \dfrac{47e^{2-\gamma}}{5760n^5}+ O\left(\dfrac{1}{n^6}\right)\approx \dfrac{e^{1.423}}{n}+ O\left(n^{-2}\right)\approx\dfrac{ 4.148655621}{n}+O\left(n^{-2}\right)} \end{array}[/math]
This means that you have a systematic error of a bit over [imath] \dfrac{3}{x} [/imath] in your formula. This becomes smaller with larger values of [imath] x. [/imath] Nevertheless it is an error. [imath] \gamma \approx 0.57721\ldots[/imath] is the Euler-Mascheroni constant, defined as [math] \displaystyle{\gamma= \lim_{n \to \infty} \left(-\log n +\sum_{k=1}^n\dfrac{1}{k}\right)}.[/math]
You should divide your formula for [imath] \pi(x) [/imath] by [imath] e^{2-\gamma} [/imath] to get an error of the magnitude [imath] 1/x^2. [/imath]

I can’t believe how fast you did these calculations. This would have taking me days, if at all. I had a feeling gamma was going to make an appearance soon. But I thought factorials would have popped up first because of the e^x and the power series. Need some time to ponder……

 

[fresh_42]

I can’t believe how fast you did these calculations. This would have taking me days, if at all. I had a feeling gamma was going to make an appearance soon. But I thought factorials would have popped up first because of the e^x and the power series. Need some time to ponder……

I looked it up on WolframAlpha, the series expansion. Modern times, you know.
https://www.wolframalpha.com/input?i=Taylor+series+of+prod+(k=2+to+n)+1/e^(1/k)++

I first tried to do it manually but that seemed to be troublesome. WA was definitely a lot faster. I have a list of links to calculators of any kind.
In case anybody is interested:
https://www.physicsforums.com/threads/list-of-online-calculators-for-math-physics-earth-and-other-curiosities.970262/