# The Naked Scientists Forum

### Author Topic: A Special Polygonal Number Counting Function.  (Read 3479 times)

#### Don Blazys

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##### A Special Polygonal Number Counting Function.
« on: 20/02/2011 06:03:42 »
Just as the "logarithmic integral" Li(x) can be used to approximate
how many prime numbers there are under a given number x,
the "Special Polygonal Number Counting Function"
which you can find here:

can be used to approximate (to a much greater degree of accuracy)
how many polygonal numbers of order greater than 2 there are
under a given number x.

Now, polygonal numbers of order greater than 2 are notoriously difficult to count,
even by computer. Thus, it is indeed very useful to have a function that
very closely approximates that count.

But what makes this function really interesting is that it requires both
the fine structure constant and the proton to electron mass ratio!

Thus, if we knew how many polygonal numbers of order greater than 2
there are under say, x=10^18, then we could immediately solve for α (alpha)
and determine the fine structure constant to a much higher degree of accuracy
than is presently known!

So, the question is, would it be possible to calculate how many
polygonal numbers of order greater than 2 there are under x=10^18 ?

Given that the number of prime numbers under x has been calculated to x=10^24,
I should think that this would be "easy", or at least "possible in a reasonable amount of time",
but as it turns out, the "coders" who determined the present "world record" which is:
ω(1,100,000,000,000)=704,398,597,754 informed me that determining ω(10^13)
would probably take them about a year, so I doubt that their machines are powerful enough
to determine ω(10^18) within our lifetimes!

However, there are institutions of higher learning that have
"supercomputers" which probably can make that calculation!

I sincerely urge them to do so and to post their results here!

What do you think?
Does anybody here in this forum have access to a "supercomputer"? [?]
I know several good coders that can help with the code.

Don.
« Last Edit: 26/02/2011 15:48:42 by Don Blazys »

#### Geezer

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##### Re: A Special Polygonal Number Counting Function.
« Reply #1 on: 21/02/2011 03:58:41 »
Given that the number of primes under x has been calculated to x=10^24, I should think that this would be "easy",
or at least "possible in a reasonable amount of time", but as it turns out, the "coders" who determined the present
"world record" ω(1,100,000,000,000)=704,398,597,754 informed me that determining ω(10^13) would probably take
about a year, so I doubt that their machines are powerful enough to determine ω(10^18)within our lifetimes!

Yes, but what's the point? Do you think we will get to the last prime number that will reveal the meaning of everything?

I once wrote a program to find prime numbers on a 1K 16-bit computer I made. The result was so boring that I made a mental note never to do anything like it again

#### Don Blazys

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• Posts: 7
##### Re: A Special Polygonal Number Counting Function.
« Reply #2 on: 21/02/2011 10:45:51 »
Quoting Geezer:
Quote
Do you think we will get to the last prime number that will
reveal the meaning of everything?

Well, there is no such thing as a "last prime number"
and even though they do get rarer as they get larger,
it is easy to prove that prime numbers go on forever.

Anyway, this topic is not about counting prime numbers
but about counting "polygonal numbers of order greater than 2".

You can find that erratic, irregular and incredible sequence here:

Quoting Geezer:
Quote
Yes, but what's the point?

The point is that a lot of time, effort and money are being spent
by scientists conducting all kinds of "physical experiments"
trying to measure the value of the fine structure constant.

The most accurate measurement so far is 1/137.035999084(51),
which means that the real value of the fine structure constant
is somewhere in between 1/137.035999135 and 1/137.035999033.

But here:

we have the only known "counting function" which actually
requires the fine structure constant in order to approximate,
to an astonishing, incredible and utterly uncanny degree of accuracy,
how many polygonal numbers of order greater than 2
there are under a given number x

Thus, it seems that if we knew how many
polygonal numbers of order greater than 2
there are under some sufficiently large number,
then we could immediately solve for α (alpha)
and determine the fine structure constant
to a much greater degree of accuracy than is presently known!

Then, if that "mathematically determined" value of
the fine structure constant were to be corroborated by
physical experiments, further physical experiments
to determine the fine structure constant would
no longer be necessary, and that would save
a lot of time effort and money!

Don

#### Don Blazys

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##### Re: A Special Polygonal Number Counting Function.
« Reply #3 on: 26/02/2011 12:29:59 »
Update:

The coder presently working on this theory with me has
just determined that ω(1,200,000,000,000)=768,434,854,386.

The function on my website:

gives B(1,200,000,000,000)*(1-α/(μ-2*e))=768,434,853,414
for a difference of -972 and a relative error of -.00000000124,
which is outstanding! An incredibly accurate prediction!

Some institution of higher learning should really be "on this like white on rice"!

Consider this...

The sequence of polygonal numbers of order greater than 2 is
"erratic", "irregular", "random" and "unpredictable".

The sequence of prime numbers is also
"erratic", "irregular", "random" and "unpredictable".

Then, since there is now evidence that the sequence of prime numbers
is intimately connected to quantum mechanics (here's just some of that evidence):

it really may not be so far fetched to suppose that the sequence of
polygonal numbers of order greater than 2 might give rise to
the Fine Structure Constant.

Maybe mother nature patterned the universe using patternless sequences!

Don.

« Last Edit: 26/02/2011 15:55:09 by Don Blazys »

#### Don Blazys

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• Posts: 7
##### A Special Polygonal Number Counting Function.
« Reply #4 on: 04/03/2011 06:56:31 »
Update:

It has just been determined that:
ω(1,300,000,000,000)=832,471,110,338.

The function on my website:

gives B(1,300,000,000,000)*(1-α/(μ-2*e))=832,471,109,826
for a difference of -512 and a relative error of -.00000000062.
Another extraordinarily accurate prediction!

Don.

#### Don Blazys

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• Posts: 7
##### A Special Polygonal Number Counting Function.
« Reply #5 on: 12/03/2011 18:35:02 »
Update:

I just received word from another coder that:

ω(10^13) = 6,403,626,146,905.

Taking this latest result, along with
the last three results in my paper:

and then solving for "alpha" in my counting function,
results in the following table:

ω(10^10) = 6,403,587,409         α = 1/137.03601728880
ω(10^11) = 64,036,148,166        α = 1/137.03593392608
ω(10^12) = 640,362,343,980       α = 1/137.03598213253
ω(10^13) = 6,403,626,146,905     α = 1/137.03603103508

Note that the above values of "α" fluctuate slightly
both above, and below the current best estimate of
the Fine Structure Constant which is:

1/137.035999084(51).

Indeed, the above four approximations of alpha average out to:

1/137.03599109564

which is extraordinarily close, considering the rather
limited amount of data that we have at our disposal!

It is also interesting that my counting function gives us
values of the Fine Structure Constant that vary,
because the latest findings by astronomers seem to indicate that
the Fine Structure Constant varies across the universe by about

You know, Polygonal Numbers of Order Greater Than 2 or
"Regular Figurative Numbers" as they are otherwise known
are every bit as basic and fundamental as the Prime Numbers.

That their counting function actually requires
both the Fine Structure Constant and
the Proton To Electron Mass Ratio
is now beyond any reasonable doubt,
and that is nothing short of astonishing!

The scientific and philosophical implications of all this...
are indeed quite profound.

For instance, can there be such a thing as "proton decay"
if the Proton to Electron Mass Ratio has actual
and eternal "mathematical underpinnings"?

If the Fine Structure Constant is, at its most fundamental level,
a "logical construct" or "mathematical object" and therefore
an eternally unavoidable consequence of logic and mathematics,
which is clearly "fine tuned" for the existence of intelligent life,
then is life not inevitable and our own continued existence,
in one form or another, not a predetermined "mathematical certainty"?

If the universe... and existence are but a manifestation of logical
and mathematical principles, then that would certainly be the case.

Don.
« Last Edit: 12/03/2011 19:02:42 by Don Blazys »

#### yor_on

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##### A Special Polygonal Number Counting Function.
« Reply #6 on: 13/03/2011 04:51:53 »
So how did you get the idea Don?
It would be immensely cool if you were right of course :)

Then we would have a indirect confirmation of constants having a 'mathematical life' of their own.
Replacing the need for physical confirmations. But? Then your idea should work equivalently well for other constants too, if we were to assume that, don't you agree? Even if not by your method, there should be equivalent ones, as I guess this is what you're thinking off?

But I agree, it's worth testing if your logic (math) is working.

#### Don Blazys

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##### A Special Polygonal Number Counting Function.
« Reply #7 on: 08/04/2011 12:30:59 »
Quoting yor_on:
Quote
So how did you get the idea Don?
It would be immensely cool if you were right of course :)

The seperation of polygonal numbers into those that
are of order greater than 2 and those that are not
is analogous to the seperation of natural numbers
into primes and composites.

Both the sequence of primes and the sequence of
polygonal numbers of order greater than 2 are
erratic, irregular, patternless, random and
utterly unpredictable, and I found it rather
amazing that mathematicians never developed
a counting function for polygonals of order
greater than 2, the way they had for the primes,
because polygonal numbers are among the most
studied numbers in the history of mathematics!

Thus it was quite natural for a curious person
such as myself to try and develop a counting
function for polygonal numbers of order > 2,
if only to see what such a function might look
like and what such a function might entail !

Now, when I began my research into this question,
the empirical data was hardly sufficient to even
hazard a guess as to what mathematical constructs
might be involved, so I developed several functions
based on what I considered to be reasonable assumptions,
employing a technique whereby one constant is allowed
to emerge as a consequence of the "relative error".

I then asked several coders to provide me with more
empirical data, and as that data came in, it became
apparent that one of the constants in one of my functions
was either the fine structure constant, or some number
very close to it.

The emergence of the fine structure constant in this
manner came as a complete and utter shock to me,
because I was expecting a mathematical rather than
a physical constant!

Quoting yor_on:
Quote
Then your idea should work equivalently well for other constants too,
if we were to assume that, don't you agree?

At this point, nobody knows.

It may turn out that the other physical constants
are related to the fine structure constant and the
proton to electron mass ratio in a way that will allow them
to be determined by some similar mathematical construct.

After all, the fine structure constant is already known
to involve many other constants, including the speed of
light and the reduced Planck constant.

Don.

#### JMLCarter

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• Posts: 143
##### A Special Polygonal Number Counting Function.
« Reply #8 on: 10/04/2011 21:02:14 »
http://donblazys.com/on_polygonal_numbers_3.pdf

Doesn't really provide enough information on your method that your results can be verified.

For some-one like me to buy in, you should clarify

the equation for β(x)                        ...since β is calculated in your table
the equation for ω(x)    (meaning omega_bar) ...since ω is calculated in your table

what values you have substituted for n and r in the polygon number generation equation (if any)
whether you have aproximatly subtracted ω from β or was it some other operation

You should declare your %err calculation as %err=Difference/ω (even though it might seem obvious to us, to others it may not be)

Until a reader could make all that add up, it does appear on the face of it that you have inserted the physical constant into your equation at the start, and got them out at the end??
Further clarification would prevent this interpretation (assuming it is in error).

You could even post the source code. tbh I would do a code walk-through/review on something of this potential importance.

#### Don Blazys

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##### A Special Polygonal Number Counting Function.
« Reply #9 on: 17/04/2011 18:33:24 »
In the function B(x), there exists one and only one
value of α which results in a "relative error" that
approaches "some constant", and given the table for ω(x),
(which was compiled not by me but by several "coders")
anyone can determine that value of α for themselves!
Thus, the value of α was never "inserted at the start"
but depends on ever increasing determinations of ω(x).

should be viewed as a "work in progress"... an invitation
for others to join in on the investigation, and especially,
as a truly worthwhile challenge to college and university
math and computer science departments, because you see,
regardless of whether or not this counting function will ultimately
result in better determinations of the Fine Structure Constant,
the fact remains that our table of ω(x) is pathetically small
when compared to existing tables of π(x), so higher counts of ω(x)
are still something that is sorely needed in our mathematical literature.

Don.
« Last Edit: 18/04/2011 00:04:10 by Don Blazys »

#### The Naked Scientists Forum

##### A Special Polygonal Number Counting Function.
« Reply #9 on: 17/04/2011 18:33:24 »