Polygonal Number Counting Function

Don Blazys · 2012-06-06 23:05:04

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

is

Integer part of is

Integer part of is
(and so on...)

The following table represents

approximated by .

______________________________________ _____________Difference
10_______________________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

as we like.

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

Don.

bobbym · 2012-06-07 00:20:34

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.

Don Blazys · 2012-06-09 23:01:25

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

? Nobody knows!
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

. That's the current "world record".
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 polygonal numbers.

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)

bobbym · 2012-06-09 23:26:42

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 ?

Bob · 2012-06-10 01:09:53

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

bobbym · 2012-06-10 01:37:16

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.

Bob · 2012-06-10 02:47:13

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

Bob

Don Blazys · 2012-06-12 19:30:43

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 without ever knowing a single prime, simply by calculating sufficiently large values of

(the number of polygonal numbers of order greater than 2 less than or equal to )
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

, and that will require
an expert coder with access to a super-computer.

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

Don.

Bob · 2012-06-12 19:44:02

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

Don Blazys · 2012-07-15 20:39:17

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 remain highly accurate is to determine higher values of .

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.

Bob · 2012-07-15 22:59:25

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

Don Blazys · 2012-07-19 00:30:33

Hi Bob,

Quoting bob bundy:

Is your generating formula

The primes are generated from

as follows:

Start with:

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

required
another century of hard labor by some of the worlds greatest mathematicians.

Now, my counting function for polygonal numbers of order greater than 2
is very, very sophisticated in that it involves not only the above prime number
counting function, but

and as well!

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 search will never end, then the fun will never end, and that's a good thing!

Don

Bob · 2012-07-19 03:56:33

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

bobbym · 2012-08-04 22:16:25

Thread closed at the request of the original poster.

Math Is Fun Forum

#1 2012-06-06 23:05:04

Polygonal Number Counting Function

#2 2012-06-07 00:20:34

Re: Polygonal Number Counting Function

#3 2012-06-09 23:01:25

Re: Polygonal Number Counting Function

#4 2012-06-09 23:26:42

Re: Polygonal Number Counting Function

#5 2012-06-10 01:09:53

Re: Polygonal Number Counting Function

#6 2012-06-10 01:37:16

Re: Polygonal Number Counting Function

#7 2012-06-10 02:47:13

Re: Polygonal Number Counting Function

#8 2012-06-12 19:30:43

Re: Polygonal Number Counting Function

#9 2012-06-12 19:44:02

Re: Polygonal Number Counting Function

#10 2012-07-15 20:39:17

Re: Polygonal Number Counting Function

#11 2012-07-15 22:59:25

Re: Polygonal Number Counting Function

#12 2012-07-19 00:30:33

Re: Polygonal Number Counting Function

#13 2012-07-19 03:56:33

Re: Polygonal Number Counting Function

#14 2012-08-04 22:16:25

Re: Polygonal Number Counting Function

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