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**Don Blazys****Member**- Registered: 2012-06-06
- Posts: 32

Let polygonal numbers of order greater than 2 be defined as the various different numbers:

which are generated by the formula:, when integers and are greater than ,and let

represent how many such numbers there are less than or equal to a given number .Then,

where:

and where:

is the "Blazys constant", which generates all of the prime numbers in sequence by the following rule:

Integer part of

isInteger part of is

Integer part of is

(

The following table represents

approximated by .______________________________________ _____________Difference10_______________________3______________________5___________________2

100______________________57_____________________60__________________3

1,000____________________622____________________628_________________6

10,000___________________6,357__________________6,364________________7

100,000__________________63,889_________________63,910_______________21

1,000,000________________639,946________________639,963______________17

10,000,000_______________6,402,325______________6,402,362_____________37

100,000,000______________64,032,121_____________64,032,273____________152

1,000,000,000____________640,349,979____________640,350,090____________111

10,000,000,000___________6,403,587,409__________6,403,587,408__________-1

100,000,000,000__________64,036,148,166_________64,036,147,620_________-546

1,000,000,000,000________640,362,343,980________640,362,340,975________-3005

10,000,000,000,000_______6,403,626,146,905______6,403,626,142,352_______-4554

100,000,000,000,000______64,036,270,046,655_____64,036,270,047,131_______476

200,000,000,000,000______128,072,542,422,652____128,072,542,422,781______129

300,000,000,000,000______192,108,815,175,881____192,108,815,178,717______2836

400,000,000,000,000______256,145,088,132,145____256,145,088,130,891_____-1254

500,000,000,000,000______320,181,361,209,667____320,181,361,208,163_____-1504

600,000,000,000,000______384,217,634,373,721____384,217,634,374,108______387

700,000,000,000,000______448,253,907,613,837____448,253,907,607,119_____-6718

800,000,000,000,000______512,290,180,895,369____512,290,180,893,137_____-2232

900,000,000,000,000______576,326,454,221,727____576,326,454,222,404______677

1,000,000,000,000,000____640,362,727,589,917____640,362,727,587,828_____-2089

Now, if we use the last 10 values of

and to solve for ,and then inject those values of into the expression: as goes to

the results will be as follows:

________

_____________________________________________________________100,000,000,000,000______64,036,270,046,655_____2.5665438294154____137.03599916477

200,000,000,000,000______128,072,542,422,652____2.5665438318173____137.03599909419

300,000,000,000,000______192,108,815,175,881____2.5665438266710____137.03599924542

400,000,000,000,000______256,145,088,132,145____2.5665438340142____137.03599902963

500,000,000,000,000______320,181,361,209,667____2.5665438339138____137.03599903258

600,000,000,000,000______384,217,634,373,721____2.5665438318063____137.03599909451

700,000,000,000,000______448,253,907,613,837____2.5665438377183____137.03599892078

800,000,000,000,000______512,290,180,895,369____2.5665438337865____137.03599903632

900,000,000,000,000______576,326,454,221,727____2.5665438317301____137.03599909675

1,000,000,000,000,000____640,362,727,589,917____2.5665438334003____137.03599904767

Taking the average of the

column results in: ,which is an exellent approximation considering that we used only

10 samples from relatively low values of ,

and taking the average of the column results in: ,

which is very close to the most precisely determined value of the fine structure constant to date,

and matches the latest Codata value perfectly!

So, in theory, if we had sufficiently large values of

, say , to about or so...then we can simply take the average of sufficiently many random samples of to get

to as many decimal places as we like,

and thereby generate the entire sequence of primes in sequential order!

It's essentially the same principle as flipping a coin sufficiently many times

and averaging out the results in order to get as close to

I really like the idea of using one erratic sequence to generate another. It's kind of like fighting fire with fire.

Don.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,242

Hi Don Blazys;

Welcome to the forum. I am familiar with some of your ideas and the controversy that surrounds them. I do not say that you are the cause of the arguing and ad hominem attacks that follow your work on other forums and blogs. If it follows you here then I must say I will moderate it strongly.

Name calling or personal attacks regardless of the reputation of the aggressor will be deleted immediately.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**Don Blazys****Member**- Registered: 2012-06-06
- Posts: 32

Thanks bobbym,

What a great name for a math forum. **Math is fun** indeed!

But why is it fun? What is it about math that makes it so enjoyable?

Well, here are several of my reasons for reveling in it.

___________________________________________Math is mysterious.______________________________________________

Everyone loves a good mystery, and math is not only one of the most important tools that scientists use in solving the riddles

and mysteries of the universe, but it is also a fascinating subject in its own right, and contains some of the most perplexing

puzzles and profound problems known to mankind.

The counting function in post #1 is an exellent example of just how mysterious some math problems can be.

How many **polygonal numbers of order greater than 2** are there less than or equal to

The above counting function can be used to approximate the answer, but the exact value of remains a mystery.

Why does approximating the number of **polygonal numbers of order greater than 2** to a high degree of accuracy

require the "running" of the **fine structure constant** which is by far the most important constant in all of physics?

Again, nobody knows! Google searching the phrase " reflexive polygons in string theory" brings up all kinds of results

showing that polygonal numbers are at the very core of string theory, but so far, that entire issue remains a mystery!

__________________________________________Math is challenging._____________________________________________

Everybody loves a challenge. Indeed, people have climbed Mt. Everest and swam across the English Channel simply

because it was a challenge and for no reason other than "it was there". A life without challenges is dull, boring and hardly

worth living while a life that is filled with challenges is extraordinarily interesting and (most importantly), loads of fun!

The counting function in post #1 is a perfect example of just how challenging some math problems can be.

Seperating the **polygonal numbers of order greater than 2** from the rest of the **polygonal numbers** is analogous to

seperating the **composite numbers** from the **prime numbers**. Both are extraordinarily hard to do, and doing either results

in sequences that are absolutely random and erratic, yet follow certain other laws in a manner that is quite predictable.

**Polygonal numbers of order greater than 2** have only been counted up to

A lot of coders tried very hard to break that record, but most of them gave up after their computers either crashed or ground

to a halt. However, I'm sure that other coders will continue trying to break that record, not only because breaking records is

a fun and challenging thing to do, but because the counting function in post #1 is perhaps the most unique counting function

in all of mathematics, and as such, gets first page ranking by Google and is even referenced in the Online Encyclopedia of

Integer Sequences. It is certainly the only counting function that involves

I put it here, just in case you might want to try and break that record.

If you don't, then please lock this thread and I will continue having fun elsewhere.

Cheers,

Don

*Last edited by Don Blazys (2012-06-09 23:04:25)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,242

Hi Don Blazys;

The name comes from the originator of the site.

I do not see any reason to close the thread.

It is interesting mathematics. Do not feel that lack of replies means lack of interest.

If you feel strongly about locking it then I will do that until such time that you request it to be opened.

In the meantime what you can do is to please rectify this error:

http://www.research.att.com/~njas/seque … &go=Search

Also, can you provide a link to verify the bound of 10^15 ?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,378

hi Don,

No, don't ask for it to be closed. I'm interested!

I must admit I was a bit put off by post #1. Too much too quickly for my little brain. I'd never even heard of polygonal numbers so I had to look it up.

http://en.wikipedia.org/wiki/Polygonal_number

for anyone else in my position.

And it's got pictures too. Those who know me, know I'm always happier when I've got a nice picture to look at.

OK. So then I moved on to your prime number generator.

That should be interesting, I thought, given that some mathematicians think it cannot be done. But I'll keep an open mind. After all, I think the aquatic ape theory is correct in the face of most scientific thinking and that humans have more than 5 senses despite what they tell you in biology text books, so why not try out this idea too.

Now I'm uncertain exactly what your generator is. Obviously, my brain is only splashing about in the wake of yours (and I'm serious, not trying to be rude I promise ) but I had a problem with this.

Integer part of 2.566543832... is 2

Integer part of 1/((2.566543832.../2)-1)=3.530176989... is 3

Integer part of is 1/((3.530176989.../3)-1)=5.658487746... is 5

(and so on...)

My problem is with that innocent "and so on"

First I tried

but that gave some negative values so I adjusted to

Better. I got 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 and I'm thinking 'hey, this is interesting!' but then it all went haywire with 40, 84, 347, 431, 479 .........

Whoops. So please would you make your generator clearer for me.

Thanks,

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,242

Hi Bob;

You are not holding enough digits. You must use 2.56654383217138884446752910633228575178297282870231464596973 and continue the calculation to at least that many digits. Then you get:

which is good up until the last entry of 148. That Blazy constant is only an approximation up there and must eventually fail.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

**Online**

**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,378

I see. I'm not sure Excel will work that accurately, but I'll give it a try.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**Don Blazys****Member**- Registered: 2012-06-06
- Posts: 32

If we know sufficiently many prime numbers in advance, then it is exeedingly easy to calculate

the Don Blazys constant to as many decimal places as we like. Here it is to 1500 digits or so:

2.5665438321713888444675291063322857517829728287023146459697335254663997198904

003462239885714780566589415300383386252694557180837585065234733899407590154521

477163056174412378465009206511654428209869679944408646919502129002995825444683

535957146252243194189226038317025371635511355609594950080639727211111880806309

433690379118715226031469192311487269910138228161615957029092483549007751626381

778170170501465893712305852748021584934680316196223087098420524922955575406332

897900513351452478128278824588603694435884921287582688488499082757951311566642

464820849280217151229993076859757596523704399063065354079256240471646093954799

424643289145352443403354672891255594682830067586909327290064450778982781780646

572326075380709000130766143755442519632323931974441018947934619264008517805956

430490179231898172371368052997230780798015735735351912474123322442624555334814

040204030157123671369216800571313500108714696094834011524274914368468088494367

975660376792450000221102311268076302327835712866173550047160050758990823559294

731332935283691934260732135205234475642016782140952781965845322346648945648788

117142343108306142383815588227207565180119949919060997313844551046494747202015

388384536230021753436402688469886081359485171994227626016304251316701623585280

851128813381229455835114685529077513922917538380128873184842938429816881693161

821371821961182096793893940762517574471742445970196513683339490300781148490252

037349719426856590001962325248818060082590913466896412315136908706594026416435

982690876451518198999891129443265858404...

Those 3 dots at the end mean that the Don Blazys constant actually has an infinite number

of digits and can therefore generate an infinite number of primes, all in sequential order.

Amazingly enough, we can * also* calculate the Don Blazys constant to as many decimal places

as we like

and solving for in the counting function in post #1.

This astonishing relationship between prime numbers and polygonal numbers of order greater than 2

really should undergo further testing using even higher values of

an expert coder with access to a super-computer.

Do you have any suggestions as to where I can find such a coder?

Don.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,378

hi Don,

I don't have the computing power to do that. But I'm very interested in why it works and I don't see why that requires any computing power at all.

So I'd still like to know the exact construction of your generating function. see post #5

Thanks,

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**Don Blazys****Member**- Registered: 2012-06-06
- Posts: 32

Quoting bob bundy:

I'm very interested in why it works and I don't see why that requires any computing power at all.

Thanks Bob.

It works only "in theory". Actually * proving* that it works may or may not be possible.

So far, all I have managed to demonstrate is that the general form of the counting function is probably correct.

However, my notebooks contain dozens of variations on that form, * all* of which are highly accurate to

and the only way to determine which of those variations will

Quoting bob bundy:

...some mathematicians think it cannot be done.

Google searching "prime number generating formulas" shows that there are many such formulas,

some of which are quite clever and interesting. The problem is that none of them are efficient

enough to be of any practical value.

Now, generating primes by counting **polygonal numbers of order greater than 2** may or may not turn out to be practical,

but again, the only way to actually test the efficiency of this method is to determine larger values of

Since this is the only known method which generates all the primes and only the primes in sequential order,

I think that testing its efficiency would be interesting, informative, and a lot of fun.

Don.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,378

hi Don,

the only way to actually test the efficiency of this method is to determine larger values .....

Trouble with this seems to me to be that this search will never end. What I was hoping for is a proof, and I don't see why that needs any computing power at all.

Let me expand on that.

There is formula

Now you could substitute values of n and show it always works for those values. In time you could try ever larger values and it would still work.

But I can show it works using algebra and that covers all values of n in one go.

Now I still have a question unanswered from post #5. I'll repeat it here:

Is your generating formula

And if not, then what?

Thanks,

Bob

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**Don Blazys****Member**- Registered: 2012-06-06
- Posts: 32

Hi Bob,

Quoting bob bundy:

Is your generating formula

The primes are generated from

as follows:Start with:

then:

then:

and so on.

Thus,

is correct.

Note that the "previous prime" is simply the **floor function** of the "previous value".

Quoting bob bundy:

What I was hoping for is a proof...

Carl Gauss discovered the "simple" **prime number** counting function:

while still in his teens. Proving that it works all the way into

requiredanother

Now, my counting function for **polygonal numbers of order greater than 2**

is very, * very* sophisticated in that it involves not only the above

counting function, but

Thus, proving its convergence with

would be extraordinarily difficult,and proving that it generates all of the primes and only the primes in sequential order would

actually require a proof of the Riemann hypothesis, which may or may not be provable!

Quoting bob bundy:

Trouble with this seems to me to be that this search will never end.

From my point of view,

.Thus, if the

Don

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,378

Thanks Don,

OK; you have fun your way and I'll have fun my way.

I don't have the computing power to maintain accuracy anyway. So I'm going to ask the question "Why does it work?" amd see where I get.

Things may go quiet for a while. I have not gone to sleep. I'm just thinking

bfn

Bob

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,242

Thread closed at the request of the original poster.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**

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