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An Exact Value For The Fine Structure Constant.


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#1 Don Blazys

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Posted 30 August 2011 - 06:48 AM

To date, by far the most accurate measurement of the fine-structure constant
(measured at the scale of the electron mass) was made by Gerald Gabrielse
and colleagues from Harvard, Cornell and RIKEN.

Measuring the "magnetic moment" of a single electron in a "quantum cyclotron" and
inserting that value into state of the art QED equations, the value they determined is:

[math]\alpha^{-1}=137.035999084(51)[/math],

which means that the fine-structure constant lies somewhere in between:

[math]\alpha^{-1}=137.035999135[/math], and

[math]\alpha^{-1}=137.035999033[/math].

Now, these values were determined back in 2008, and since then,
no significant improvement in accuracy was ever accomplished,
despite enormous improvements in both the design of the equipment
and the QED equations themselves.

Thus, many scientists now suspect that further refinements in
the value of the fine-structure constant may not even be possible,
and that the last two values represent, for all practical purposes,
the actual lower and upper bounds of the fine-structure constant as
measured at the scale of the electron mass. Even Gabrielse himself
believes that the above values will hold for a long, long time to come.

Given the above facts, it now seems that in order to be correct,
any mathematical expression which results in the fine-structure constant
must not only match the above experimental value exactly, but must also include,
within it's form, some simple way of expressing, with the same degree of absolute accuracy,
those seemingly inherent lower and upper bounds.

Now, some of you may remember this article: http://donblazys.com...l_numbers_3.pdf
which describes a finding that occured right here at Hypography many moons ago.
Well, thanks to a great fellow named Lars Blomberg, (who found it via the OEIS)
we now have values of [math]\varpi(x)[/math] to [math]x=10^{15}[/math].

This information was crucial in not only greatly improving the "counting function",
but also allowed me to derive these values of the fine-structure constant as well:

[math]\alpha^{-1}=137.035999084=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{1}{(6*\pi^{5}*e^{2}-2*e^{2})}[/math]

[math]\alpha^{-1}=137.035999135=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{1}{(6*\pi^{5}*e^{2}-2*e^{1})}[/math]

[math]\alpha^{-1}=137.035999033=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{1}{(\mu*e^{2}-2*e^{\frac{5}{2}})}[/math]

where [math]\mu=1836.15267247(80)[/math] is the "proton to electron mass ratio", and
[math]A=2.566543832171388844467529...[/math], is that very special "Blazys Constant"
which generates all of the prime numbers, in sequential order, by the following simple method:

Note that the whole number part is the first prime [math]2[/math], and that:

[math]((2.566543832171388844467529...)/2-1)^{-1}[/math]

is approximately:[math](3.530176989721365539402422...)[/math],

where the whole number part is the second prime [math]3[/math], and that:

[math]((3.530176989721365539402422...)/3-1)^{-1}[/math]

is approximately [math](5.658487746849688216649061...)[/math],

where the whole number part is the third prime [math]5[/math], and so on.

(In short, we divide the approximate number by it's whole number part, subtract [math]1[/math],
and take the reciprocal of the result to get the next approximate number whose whole number part is the next prime!)

That the fine-structure constant is thus related to the prime numbers was also discovered (independently) by Ke Xiao
who publised his findings in a paper entitled "Dimensionless Constants and Blackbody Radiation Laws" in
The Electronic Journal of Theoretical Physics. (It can be "Googled".)

I will post Lars Blomberg's determinations of [math]\varpi(x)[/math] in my next post,
and a revised "polygonal number counting function" in the post after that.

There's more... a lot more... but that's all for now.

It's good to be back.

Don.
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#2 CraigD

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Posted 31 August 2011 - 10:04 AM

It's good to be back.

It’s good to see you’re still voraciously playing with finding expressions for physical constants (if [math]\alpha[/math] actual is a constant, which isn’t certain – see its wikipedia article for more).

To date, by far the most accurate measurement of the fine-structure constant (measured at the scale of the electron mass) was made by Gerald Gabrielse and colleagues from Harvard, Cornell and RIKEN.

Measuring the "magnetic moment" of a single electron in a "quantum cyclotron" and inserting that value into state of the art QED equations, the value they determined is:

[math]\alpha^{-1}=137.035999084(51)[/math],

Point of accuract: I understand from the above linked wikipedia article that a slightly more precise recommended value has been calculated from measurements of the other physical constants defining
[math]\alpha^{-1} = \frac{2 \varepsilon_0 h c}{e^2} = 137.035999074(44)[/math]

Now, these values were determined back in 2008, and since then, no significant improvement in accuracy was ever accomplished, despite enormous improvements in both the design of the equipment and the QED equations themselves.

Thus, many scientists now suspect that further refinements in the value of the fine-structure constant may not even be possible, and that the last two values represent, for all practical purposes, the actual lower and upper bounds of the fine-structure constant as measured at the scale of the electron mass. Even Gabrielse himself believes that the above values will hold for a long, long time to come.

Do you have a source for this, Don :QuestionM

My read of "New Measurement of the Electron Magnetic Moment and the Fine Structure Constant" (2008 D. Hanneke, S. Fogwell, G. Gabrielse) finds that the authors note the new [imath]\alpha[/imath] estimate is 2.7 times more accurate than the 2006 one it supersedes, supporting new physics experiments, but that like those that came before it, it can be improved with improved instrumentation and numeric approximation methods.

Given the above facts, it now seems that in order to be correct, any mathematical expression which results in the fine-structure constant must not only match the above experimental value exactly, but must also include, within it's form, some simple way of expressing, with the same degree of absolute accuracy, those seemingly inherent lower and upper bounds.

For the reasons I gave above, I don’t think this is true. Hanneke, Fogwell, and Gabrielse’s 2008 approximation is an approximate, not an exact value. Its error bounds are also approximate, and would be different if different statistical levels of confidence were assumed, or if they had simply collected and included more data in their analysis before publishing it.

[math]\alpha^{-1}=137.035999084=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{1}{(6*\pi^{5}*e^{2}-2*e^{2})}[/math]

[math]\alpha^{-1}=137.035999135=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{1}{(6*\pi^{5}*e^{2}-2*e^{1})}[/math]

[math]\alpha^{-1}=137.035999033=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{1}{(\mu*e^{2}-2*e^{\frac{5}{2}})}[/math]

where [math]\mu=1836.15267247(80)[/math] is the "proton to electron mass ratio" ...

Cool and fun expressions, but don’t think you should use the “exactly equals” symbol (“=”) rather than an “approximately equals” symbol (such as [imath]\dot=[/imath]). Unless you ad-hock define A to be the reciprocal of the rest of the expression times the given rational [imath]\alpha^{-1}[/imath] value, the expressions to the rightmost equal sign in those equations give irrational numbers with no exact finite length decimal number representation. As you note, You’ve also included a physical constant, [imath]\mu[/imath]

... and [math]A=2.566543832171388844467529...[/math], is that very special "Blazys Constant"
which generates all of the prime numbers, in sequential order, by the following simple method ...

I don’t think you should claim that the integer parts of
[math]A_n = \frac1{\frac{A_{n-1}}{\lfloor A_{n-1} \rfloor}-1}[/math]
give “all of the prime numbers” for some [math]A_1[/math]. As we showed in 2008 in the thread The Holy Grail Of Mathematics, no [imath]A_1[/imath] can generate more than the first 17 primes using this formula – as you put it, it’s “just a really wierd and incredible coincidence”, a “mere ‘curio’”.

It’s fun to be rereading and kicking this stuff around again, though. :)
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#3 Don Blazys

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Posted 04 September 2011 - 08:09 AM

Quoting CraigD:

It’s good to see you’re still voraciously playing with finding expressions for physical constants.


Unfortunately for me, "voracious playing" is more for younger folks, such as yourself,
who are still "full of beans" and have lots of "nervous energy" to spare. :)

Old guys like me just sit back and let the solutions come while playing with our
cute little grandchildren.

Besides, like I said before, I have very little interest in "physical constants",
and my main goal for now is to continue to investigate and develop
counting functions for polygonal numbers of order greater than 2.

That is something that is sorely needed in our mathematical literature,
and mine is the very first, and still the only one! :)

The extraordinarily accurate formula for [math]\alpha^{-1}[/math] that I have presented here came about solely as
a by-product of my research into counting functions for higher order polygonals.
(The similarity between it and the counting function on my website is quite apparent.)

...if [math]\alpha[/math] actually is a constant, which isn’t certain...


For me, the important thing is that [math]\alpha[/math] has a particular value when measured at a particular energy.

Measured at the "low energy" scale of the electron mass, it's value is [math]\alpha^{-1}=137.035999084(51)[/math].

Measured at the "high energy" scale of the W boson, its value is about [math]\alpha^{-1}=128[/math],

It's value at "zero energy" is really the biggest mystery of all, because that can't be measured,
but only deduced from theory. However, most theorists do agree that at zero energy,
the fine structure constant is somewhere around [math]\alpha^{-1}=137.03604[/math] or [math]\alpha^{-1}=137.03605[/math].
(Los Alamos National Laboratories uses the latter in all their calculations, while Wikipedia
doesn't even bother with the last two decimal places and gives [math]\alpha^{-1}=137.036[/math].)

Now, the most recently determined values of [math]\varpi(x)[/math], as determined by Lars Blomberg, are as follows:

[math]\varpi(10^{13})=6,403,626,146,905[/math]

[math]\varpi(10^{14})=64,036,270,046,655[/math]

[math]\varpi(10^{15})=640,362,727,589,917[/math]

and if we now use the above values to solve for [math]\alpha^{-1}[/math] in the counting function,
which for your convenience, can be found here: http://donblazys.com...l_numbers_3.pdf ,
we get, respectively:

[math]\alpha^{-1}=137.03603021541[/math].

[math]\alpha^{-1}=137.03604279074[/math].

[math]\alpha^{-1}=137.03604679116[/math].

Could it be that the counting function will give us that most mysterious number of all,
the fine-structure constant at zero energy, provided that we are able to determine sufficiently high
values of [math]\varpi(x)[/math]? Yes! It could be! It sure seems to be settling in that vicinity!

This is even better and more exiting than I ever thought possible, and here's the "icing on the cake"...

The value of [math]\alpha^{-1}[/math], as measured at the "high energy" scale of the W boson, is:

[math] \alpha^{-1}=128.08370052236=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{e^{\sqrt(137.035999084)}}{(6*\pi^{5}*e^{2}-2*e^{2})} [/math]

where [math]A=2.566543832171388844467529[/math] and [math]137.035999084[/math] is the "low energy"
value of [math]\alpha^{-1}[/math].

Quoting CraigD:

Point of accuract: I understand from the above linked wikipedia article that
a slightly more precise recommended value has been calculated from measurements of
the other physical constants defining [math] \alpha^{-1} = \frac{2 \varepsilon_0 h c}{e^2} = 137.035999074(44) [/math]


That "Codata" value is not from any particular experiment, but comes from multiple sources.
Most of its "weight" is still from the Gabrielse experiments, but it is also "tainted" by
other, much less reliable results. The last "Codata" value of [math]\alpha[/math], was off by more than
6 standard deviations, which is huge!

Quoting CraigD:

I don’t think you should claim that the integer parts of
[math] A_n = \frac1{\frac{A_{n-1}}{\lfloor A_{n-1} \rfloor}-1} [/math]

give “all of the prime numbers” for some .


Of course, I'm not claiming any such thing. At this point in time,
there is simply not enough information to know if the actual Blazys constant:
[math]2.566543832171388844467529...[/math] is involved or not.

However, just because the "Holy Grail" thread turned out to involve
an incredible approximation to the Blazys Constant
rather than the Blazys Constant itself, it doesn't mean that
we should assume anything one way or the other in this particular case,
which is completely new, and totally different.

It's a pretty "cut and dry" proposition.

Either [math]\alpha^{-1}[/math] will appear as [math]137.035999083778...[/math]

or it won't.

Don.

#4 LBg

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Posted 15 September 2011 - 04:36 AM

Yes, I am the fellow that helped Don calculate w(10^15).

I have extended the calculations in Don's paper, and made some investigations on my own.

The results indicate that the presence of the "fine structure constant" in these data is rather speculative.
It is even doubtful that the form of the equation is the best one for approximating the data.

I have attached my investigations in PDF form.

/Lars

Attached Files


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#5 Turtle

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Posted 15 September 2011 - 09:21 PM

Yes, I am the fellow that helped Don calculate w(10^15).


a fine 1015 kudos on you lb for for your willingness & workmanship!! :thumbs_up :xparty:

I have attached my investigations in PDF form.

/Lars


may i quote from your tables in our thread on non-figurate (non-polygonal) numbers? thnx again and welcome to the hypography menagerie. :turtle:

#6 Rade

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Posted 16 September 2011 - 12:45 AM

From this link, the fine structure constant "error bars" may not be errors at all. The so-called constant may not be a constant but have different values at different times and different places in the universe:

http://www.physorg.c...s202921592.html

#7 Don Blazys

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Posted 24 September 2011 - 09:10 AM

As we have seen, for relatively low values of [math]x[/math], the function on my website gives exellent approximations of [math]\varpi(x)[/math]
when the inverse fine-structure constant [math]\alpha^{-1}[/math] assumes the "low energy" value: [math]\alpha^{-1}=137.035999084(51)[/math].

However, in order to maintain the uncanny accuracy of this counting function for very high values of [math]x[/math], (say, above [math]x=10^{12}[/math]),
we must remember that the fine-structure constant is actually a "running constant", and that it's "zero energy" value is [math]\alpha^{-1}\approx 137.03605(5)[/math],
which means that it can be viewed as a function whose range is [math]\alpha^{-1}\approx137.036[/math] to [math]\alpha^{-1}\approx137.0361[/math].

("Google searching" the number [math]137.03604(11)[/math] shows that this is by far the most widely used and
commonly accepted value of the fine structure constant. Incredibly, the range required by my counting function
falls exactly within that range, which is [math]\alpha^{-1}\approx137.03593[/math] to [math]\alpha^{-1}\approx137.03615[/math].)

The following polygonal number counting function is basically a simplified version of the one on my website:


[math]\varpi(x)\approx\left(\left(\sqrt{\left(\left(1-\frac{1}{\left(\alpha*\pi*e+e\right)}\right)*x\right)}-\frac{1}{4}\right)^{2}-\frac{1}{16}\right)*\left(1-\frac{\alpha}{\left(6*\pi^{5}-2*e\right)}\right)[/math]

where

[math]\alpha=\left(\left(A^{-1}*\pi*e+e\right)*\left(\pi^{e}+e^{\left(\frac{-\pi}{2}\right)}\right)-\frac{\left(\left(\pi^{e}+4+\frac{1}{16}\right)*\left(\ln\left(x\right)\right)^{-1}-\left(\sqrt{2}-1\right)\right)}{\left(6*\pi^{5}*e^{2}-2*e^{2}\right)}\right)^{-1}[/math]

and

[math]A=2.566543832171388844467529...[/math]

As you can see, the only real difference is that [math]\alpha[/math] is now being applied as the "running constant" that it actually is!

The results are stunning, and I will post them when I have more time.

Don.

#8 Don Blazys

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Posted 25 September 2011 - 01:08 AM

[math]x[/math]_______________________[math]\varpi(x)[/math]_________________[math]B(x_{F\alpha})[/math]__________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_________64,036,147,619_________-547
1,000,000,000,000________640,362,343,980________640,362,340,964________-3016
10,000,000,000,000_______6,403,626,146,905______6,403,626,142,294_______-4611
100,000,000,000,000______64,036,270,046,655_____64,036,270,047,024_______369
200,000,000,000,000______128,072,542,422,652____128,072,542,422,825______173
300,000,000,000,000______192,108,815,175,881____192,108,815,179,004______3123
400,000,000,000,000______256,145,088,132,145____256,145,088,131,479_____-668
500,000,000,000,000______320,181,361,209,667____320,181,361,209,090_____-577
600,000,000,000,000______384,217,634,373,721____384,217,634,375,409______1688
700,000,000,000,000______448,253,907,613,837____448,253,907,608,821_____-5016
800,000,000,000,000______512,290,180,895,369____512,290,180,895,262_____-107
900,000,000,000,000______576,326,454,221,727____576,326,454,224,974______3247
1,000,000,000,000,000____640,362,727,589,917____640,362,727,590,859______942

#9 LBg

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Posted 25 September 2011 - 03:16 AM

As we have seen, for relatively low values of [math]x[/math], the function on my website gives exellent approximations of [math]\varpi(x)[/math]
when the inverse fine-structure constant [math]\alpha^{-1}[/math] assumes the "low energy" value: [math]\alpha^{-1}=137.035999084(51)[/math].

However, in order to maintain the uncanny accuracy of this counting function for very high values of [math]x[/math], (say, above [math]x=10^{12}[/math]),
we must remember that the fine-structure constant is actually a "running constant", and that it's "zero energy" value is [math]\alpha^{-1}\approx 137.03605(5)[/math],
which means that it can be viewed as a function whose range is [math]\alpha^{-1}\approx137.036[/math] to [math]\alpha^{-1}\approx137.0361[/math].


The initial hypothesis was that the counting function could be used to give a value of [math]\alpha^{-1}=137.035999084(51)[/math] which would become accurate as more data from the counting was obtained.

When it turns out that more counting data does not support this hypothesis it is abandoned and another one is constructed, using the concept of a "running constant". I don't see why [math]\alpha^{-1}=137.035999084(51)[/math] which is a number with a stated uncertainty should be regarded as a true constant whereas [math]\alpha^{-1}\approx 137.03605(5)[/math] which is another number with a stated uncertainty should be called "running". In any case, a given formula connecting the counting data with [math]\alpha^{-1}[/math] can only give a single value for [math]\alpha^{-1}[/math], not a range of values.

The following polygonal number counting function is basically a simplified version of the one on my website:


It is not a simplified version, it is more complicated in that [math]x[/math] is now included in the expression for [math]\alpha[/math].

The results are stunning, and I will post them when I have more time.


Yes, you are right, the new function matches the current counting data better.

/LBg

#10 Don Blazys

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Posted 25 September 2011 - 07:26 AM

Here:

http://www.integralw.../piacenza4.html

is a wonderful article about the fine structure constant that mentions the outstanding work of Professor Jose Alvarez Lopez.
His theory that all non-dimentional numbers can be determined by the number [math]137.03604[/math] is truly profound,
but what I found really interesting was this:

Quoting the linked article:

Professor Alvarez Lopez liked to point out that the existence of powers of 2 are common in atomic physics,
but told me that finding powers of 10, in what seem to be fundamental numbers for this discipline,
is "outstanding, previously unknown in nature and quite mysterious for orthodoxy".


Well, the table in my previous post shows that roughly the first half of the digits in every consecutive power of 10 are repeated and therefore
easily predictable. That's comparable to the prime counting function [math]Li(x)[/math], which predicts [math]\pi(x)[/math] to about the same degree of accuracy.


Quoting LBg:

The initial hypothesis was that the counting function could be used to give a value of [math]137.035999084(51)[/math]
which would become accurate as more data from the counting was obtained.


That hypothesis was made when the known values of [math]\varpi(x)[/math] were much, much, much less than they are now.
However, mathematics is full of surprises, and in the process of trying to verify that hypothesis, it was discovered that the expression:

[math](A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{q}{(6*\pi^{5}*e^{2}-2*e^{2})} [/math]

gives us all the significant values of the "fine-structure constant", provided that we choose proper values of [math]q[/math].

For instance, if we choose [math]q=1[/math], then the result is indeed [math]\alpha^{-1}=137.035999084[/math],
which matches Professor Gabrielse's results with uncanny precision! Then again...

Choosing [math]q=(\sqrt{2}+1)^{-1}[/math] results in [math]\alpha^{-1}=137.03604230759[/math], which is also significant in that it works very, very well
in my polygonal number counting function and falls right in the range of the best theoretical "zero energy" values of [math]\alpha[/math]. Then, yet again...

choosing [math]q=e^{\sqrt{137.03604...}}[/math] results in [math]\alpha^{-1}=128.08[/math] which is also very significant in that it is
exactly the value of the fine structure constant measured at the high energy value of the W boson!

Thus, what we found is a true "fine structure formula", that is also absolutely necessary if we are to
"make the math work" and closely approximate the purely mathematical function [math]\varpi(x)[/math].

Moreover, the counting function is now so good that it shows that the "random fluctuations" in [math]\varpi(x)[/math]
are either increasing very, very slowly, or not at all.

Best of all, the "fine structure formula" seems to involve the prime numbers via the "Blazys constant",
which may (or may not) lead to some construct that generates them.

Like I said, math is full of surprises, and you just never know...

Regardless of whether or not you are satisfied with the progress that has been made,
my sincere thanks goes out to You, Turtle, Phillip and everyone else who joined in and helped.

In my most humble opinion, the work that we all put in was well worth the effort.

Don.

#11 Don Blazys

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Posted 30 October 2011 - 07:35 AM

Quoting Rade:

From this link, the fine structure constant "error bars"
may not be errors at all. The so-called constant may not be
a constant but have different values at different times and
different places in the universe:

http://www.physorg.c...s202921592.html


I agree... and isn't it strange how my truly accurate counting function for
polygonal numbers of order greater than 2 actually requires some
function with a range of about [math]\alpha^{-1}=137.03604(11)[/math]?

Think about it. A true "theory of everything" would have to include values of
the fine structure constant measured at all energy levels and at all times
and places in the universe.

Thus, the purely mathematical model of the fine structure constant
that is slowly emerging as the result of our ongoing efforts to develop
the most accurate counting function possible for polygonal numbers
of order greater than 2
is interesting indeed, because it encompasses
all possible values of the fine structure constant and not just those
measured at or around the low energy value of the electron mass.

Don

#12 Don Blazys

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Posted 03 November 2011 - 07:16 AM

Quoting LBg:

It is even doubtful that the form of the equation
is the best one for approximating the data.


The form of the equation that I developed is correct.

However, the existing determinations of [math]\varpi(x)[/math] are still insufficient
for us to determine which constants are actually involved.

To illustrate, here is a counting function of the exact same form
as that in post #7, but which contains a few different constants...

[math] \varpi(x)\approx\left(\left(\sqrt{\left(\left(1-\frac{1}{\left(\alpha*\pi*e+e\right)}\right)*x\right)}-\frac{1}{4}\right)^{2}-\frac{1}{16}\right)*\left(1-\frac{\alpha}{\left(6*\pi^{5}-2*e\right)}\right) [/math]

where:

[math] \alpha=\left(\left(A^{-1}*\pi*e+e\right)*\left(\pi^{e}+e^{\left(\frac{-\pi}{2}\right)}\right)-\frac{\left(\left(\pi^e+\frac{\pi^{2}}{6}+\sqrt{2}+1\right)*\left(\ln\left(x\right)\right)^{-1}-\left(\sqrt{2}-1\right)\right)}{\left(6*\pi^{5}*e^{2}-2*e^{2}\right)}\right)^{-1} [/math]

which results in the following table:

[math]x[/math]_______________________[math]\varpi(x)[/math]_________________ [math] B(x_{F\alpha}) [/math]_________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_________64,036,147,619_________-547
1,000,000,000,000________640,362,343,980________640,362,340,963________-3017
10,000,000,000,000_______6,403,626,146,905______6,403,626,142,289_______-4616
100,000,000,000,000______64,036,270,046,655_____64,036,270,046,979_______324
200,000,000,000,000______128,072,542,422,652____128,072,542,422,737______85
300,000,000,000,000______192,108,815,175,881____192,108,815,178,873______2992
400,000,000,000,000______256,145,088,132,145____256,145,088,131,304_____-841
500,000,000,000,000______320,181,361,209,667____320,181,361,208,875_____-792
600,000,000,000,000______384,217,634,373,721____384,217,634,375,152______1431
700,000,000,000,000______448,253,907,613,837____448,253,907,608,523_____-5314
800,000,000,000,000______512,290,180,895,369____512,290,180,894,923_____-446
900,000,000,000,000______576,326,454,221,727____576,326,454,224,594______2867
1,000,000,000,000,000____640,362,727,589,917____640,362,727,590,438______521

Here's another:

[math] \varpi(x)\approx\left(\left(\sqrt{\left(\left(1-\frac{1}{\left(\alpha*\pi*e+e\right)}\right)*x\right)}-\frac{1}{4}\right)^{2}-\frac{1}{16}\right)*\left(1-\frac{\alpha}{\left(6*\pi^{5}-2*e\right)}\right) [/math]

where:

[math] \alpha=\left(\left(A^{-1}*\pi*e+e\right)*\left(\pi^{e}+e^{\left(\frac{-\pi}{2}\right)}\right)-\frac{\left(\left(e^{\pi}+\pi+\sqrt{5}-2\right)*\left(\ln\left(x\right)\right)^{-1}-\left(\sqrt{2}-1\right)\right)}{\left(6*\pi^{5}*e^{2}-2*e^{2}\right)}\right)^{-1} [/math]

which results in the following table:

[math]x[/math]_______________________[math]\varpi(x)[/math]_________________ [math] B(x_{F\alpha}) [/math]_________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_________64,036,147,619_________-547
1,000,000,000,000________640,362,343,980________640,362,340,963________-3017
10,000,000,000,000_______6,403,626,146,905______6,403,626,142,289_______-4616
100,000,000,000,000______64,036,270,046,655_____64,036,270,046,979_______324
200,000,000,000,000______128,072,542,422,652____128,072,542,422,738______86
300,000,000,000,000______192,108,815,175,881____192,108,815,178,875______2994
400,000,000,000,000______256,145,088,132,145____256,145,088,131,306_____-839
500,000,000,000,000______320,181,361,209,667____320,181,361,208,878_____-789
600,000,000,000,000______384,217,634,373,721____384,217,634,375,156______1435
700,000,000,000,000______448,253,907,613,837____448,253,907,608,528_____-5309
800,000,000,000,000______512,290,180,895,369____512,290,180,894,928_____-441
900,000,000,000,000______576,326,454,221,727____576,326,454,224,599______2872
1,000,000,000,000,000____640,362,727,589,917____640,362,727,590,444______527

And here is yet another variation on that form which I think is exeptionally beautiful...

[math] \varpi(x)\approx\left(\left(\sqrt{\left(\left(1-\frac{1}{\left(\alpha*\pi*e+e\right)}\right)*x\right)}-\frac{1}{4}\right)^{2}-\frac{1}{16}\right)*\left(1-\frac{\alpha}{\left(6*\pi^{5}-e^{2}\right)}\right) [/math]

where:

[math] \alpha=\left(\left(A^{-1}*\pi*e+e\right)*\left(\pi^{e}+e^{\left(\frac{-\pi}{2}\right)}\right)-\frac{\left(\left(\pi^e+e^{\frac{-\pi}{2}}+4+\frac{1}{16}\right)*\left(\ln\left(x\right)\right)^{-1}+e^{\frac{-\pi}{2}}\right)}{\left(6*\pi^{5}*e^{2}-2*e^{2}\right)}\right)^{-1} [/math]

and which results in the following table:

[math]x[/math]_______________________[math]\varpi(x)[/math]_________________ [math] B(x_{F\alpha}) [/math]_________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________640,362,340,970________-3010
10,000,000,000,000_______6,403,626,146,905______6,403,626,142,331_______-4574
100,000,000,000,000______64,036,270,046,655_____64,036,270,047,185_______530
200,000,000,000,000______128,072,542,422,652____128,072,542,423,029______377
300,000,000,000,000______192,108,815,175,881____192,108,815,179,210______3329
400,000,000,000,000______256,145,088,132,145____256,145,088,131,659_____-486
500,000,000,000,000______320,181,361,209,667____320,181,361,209,229_____-438
600,000,000,000,000______384,217,634,373,721____384,217,634,375,490______1769
700,000,000,000,000______448,253,907,613,837____448,253,907,608,833_____-5005
800,000,000,000,000______512,290,180,895,369____512,290,180,895,193_____-176
900,000,000,000,000______576,326,454,221,727____576,326,454,224,816______3089
1,000,000,000,000,000____640,362,727,589,917____640,362,727,590,603______686

Or how about:

[math] \varpi(x)\approx\left(\left(\sqrt{\left(\left(1-\frac{1}{\left(\alpha*\pi*e+e\right)}\right)*x\right)}-\frac{1}{4}\right)^{2}-\frac{1}{16}\right)*\left(1-\frac{\alpha}{\left(6*\pi^{5}-e^{2}\right)}\right) [/math]

where:

[math] \alpha=\left(\left(A^{-1}*\pi*e+e\right)*\left(\pi^{e}+e^{\left(\frac{-\pi}{2}\right)}\right)-\frac{\left(\left(5^2+\sqrt{3}\right)*\left(\ln\left(x\right)\right)^{-1}+e^{\frac{-\pi}{2}}\right)}{\left(6*\pi^{5}*e^{2}-2*e^{2}\right)}\right)^{-1} [/math]

which results in:

[math]x[/math]_______________________[math]\varpi(x)[/math]_________________ [math] B(x_{F\alpha}) [/math]_________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________640,362,340,971________-3009
10,000,000,000,000_______6,403,626,146,905______6,403,626,142,335_______-4570
100,000,000,000,000______64,036,270,046,655_____64,036,270,047,218_______563
200,000,000,000,000______128,072,542,422,652____128,072,542,423,096______444
300,000,000,000,000______192,108,815,175,881____192,108,815,179,308______3427
400,000,000,000,000______256,145,088,132,145____256,145,088,131,789_____-356
500,000,000,000,000______320,181,361,209,667____320,181,361,209,390_____-277
600,000,000,000,000______384,217,634,373,721____384,217,634,375,683______1962
700,000,000,000,000______448,253,907,613,837____448,253,907,609,056_____-4781
800,000,000,000,000______512,290,180,895,369____512,290,180,895,448______79
900,000,000,000,000______576,326,454,221,727____576,326,454,225,101______3374
1,000,000,000,000,000____640,362,727,589,917____640,362,727,590,919______1002

You see, all of the above counting functions give exellent approximations of [math]\varpi(x)[/math] up to [math]x=10^{15}[/math],
so the only way to determine which constants are actually involved is to determine yet higher values of [math]\varpi(x)[/math],
and while it may be beyond the abilities of todays "coders" to determine, say, [math]\varpi(10^{16})[/math], at least now,
we can all be absolutely certain that any truly accurate counting function for polygonal numbers of
order greater than 2 must involve some function whose range is that of the fine structure constant.

In other words, without the fine structure constant, there can be no truly accurate counting function for
polygonal numbers of order greater than 2.

Don

#13 phillip1882

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Posted 03 November 2011 - 03:06 PM

i dont see why it has to be so complicated. for example, the very simple equation...
when x1 = 100,000,000 y1 = 64,032,121
when x2 = 1,000,000,000,000 y2 = 640,362,343,980
x2 -x1 = 999,900,000,000
y2 -y1 = 640,298,311,859
m = 0.64036234809380938093809380938094
64,032,121 = m*100,000,000 +b
b = -4113

y = 0.64262348*x -4113
check...

x = 10,000,000,000
aprox = 6,403,619,368
actual = 6,403,587,409
differance = 31,958
error .00000499
and you can do even better with quadratic rather than linear.

#14 Don Blazys

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Posted 08 November 2011 - 07:37 AM

Quoting phillip 1882:

I dont see why it has to be so complicated.


To me, it doesn't appear complicated at all!

Moreover, the accuracy of my counting function is stunning,
and the relationships it employs are quite profound.

Frankly, I'm surprised that you can't see that.

Let's compare results...

Quoting phillip's results:

actual = 6,403,587,409

approx = 6,403,619,368

difference = 31,958


Quoting my results:

actual = 6,403,587,409

approx = 6,403,587,408

difference = -1


I leave it to you to decide which function is closer to the truth.

Here, at Hypography, we have a chance to finish what is
probably the most amazing counting function ever devised
for polygonal numbers of order greater than 2.

It's something that has never been done in the entire history of mathematics,
and it's something that our mathematical literature desperately needs.

And all we need are higher values of [math]\varpi(x)[/math].

Don.

#15 Don Blazys

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Posted 03 December 2011 - 10:29 PM

Quoting LBg:

Yes, you are right, the new function matches the current counting data better.


Of course I'm right!

You and phillip 1882 may be great coders, but my counting function for polygonal numbers
of order greater than 2, which does involve the fine structure constant in a most profound way,
is infinitely superior to the "counting functions" that you and phillip 1882 came up with.

In fact, an in-depth analysis of my counting function now proves that the entire range of
the fine structure constant's domain
absolutely and unequivocally must be involved in
any similar polygonal number counting function that approaches mine in accuracy.

Think about that for a while, and allow your mind to grasp the extraordinary significance of
what that actually means.

It means that for the first time in the history of science, we have demonstrated conclusively
and beyond any reasonable doubt that both [math]\alpha[/math] and [math]\mu[/math] , the two dimensionless, so called
"physical" constants of elementary particle physics, are in fact, mathematical constants,
which are absolutely required in any polygonal number "density function" whose over-all
relative error continuously decreases, and becomes [math]0[/math] an infinite number of times.

And since the fine structure constant involves not only [math]\pi[/math] and [math]e[/math], but the entire sequence of
primes as well, it also means that, at least in both theory and principle, the entire sequence of
prime numbers can be generated simply by determining ever higher values of [math]\varpi(x)[/math]
and solving for [math]A=2.566543832171388844467529...[/math]....

And of course, the good news is that this can easily be tested, but what really makes this
entire theory so darn tantalizing is that [math]\varpi(x)[/math] becomes more and more linear
(and in that sense, more and more predictable) as [math]x[/math] increases. In other words,
as [math]x[/math] increases into infinity, [math]\varpi(x)[/math] becomes a "virtual" straight line.

That's more good news indeed..., but here's some really great news...

If you Google search the phrase "fine structure constant, exact value", then you will find that
Google now ranks this thread at #1, just as it ranks all three versions of my incredible
"Proof of Beal's Conjecture" at the very top! (Perhaps that's why I am now getting all those
invitations from various science forums to join them, and various women to marry them!)

Anyway, as I will probably be too busy to do much posting during the coming holiday season,
I want to take this opportunity to thank you all for making this thread possible, and to wish you all
a merry Christmas and happy New Year!

God bless you all,

Don.

#16 Don Blazys

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Posted 16 January 2012 - 10:25 PM

Well, I hope everyone here had a great Christmas vacation!

I sure did.

Now, I have some more really good news for you!

As you all know, the subject of Polygonal numbers is huge,
and constitutes an extraordinarily important part of number theory.

Thus, every truly smart Hypographer should be overjoyed to learn that "Google searching"
the words polygonal numbers now brings up this counting function on... get this...
the very first page!!!

Isn't that great? Doesn't that make you happy?

Don.
  • JMJones0424 likes this

#17 pascal

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Posted 30 January 2012 - 11:57 PM

Hi folks. I just signed up for this forum. Interesting discussion. Two days ago I pondered the following:

137/128 equals, approximately, 173/162. Here of course are the (whole number reciprocal approximation) low and high energy Bohr atom FSC's on the left, and the supposedly more realistic neutral atom limit for the right numerator. I don't know whether the actual and calculated values make this better or worse. Its only been 2 days!

Now, what is interesting here is that the denominators are both half and double squares (64-128-256) and (81-162-324). 162 shows up on quite a few phenomena I've been exploring related to atomic physics. For those of you fascinated by how powers of ten seem to interact with things note that it is 100x the rounded value of Phi, the Golden Ratio. But 162 can be generated as a string of Phi-related numbers just by interactions with Fibonacci and Lucas numbers (likely also by higher analogues): ...38=34+3, 62=55+7, 100=89+11, 162=144+18, 262=233+29....

162 is also the internal angle of the 20-gon, again relating it (in a 360 degree circle) to the Golden Ratio trigonometrically, and also relating it to 144 (the internal angle of a decagon, similarly Phi-motivated), and a whole slew of other angles (such as 108 for the pentagon) also relating through + or - 2x their sines or cosines, all separated from each other by 18 degrees. So there is a polygonal relationship as well.

Further, if you look at 137 and 173, their difference is 36 (twice 18), with an average value of 155- this latter number just happens also to be 1/4 of 620, relating it to 1000x the rounded value of the lower Golden Ratio. It turns out that several important numbers in the atomic realm are different from 155 just by using Lucas or doubled Lucas numbers (the denominators 128 and 162 differ by 34, a Fibonacci number)- for example 155+29=184, which is a nuclear magic number. The prior nuclear magic number of 126 is 155-29. And so on. You hit all sorts of things this way. Even data from the Miley tabletop nuclear experiments generate these numbers for peaks and valleys of nuclide production (ignore this if you don't believe in LENR).

All these observations (there are many) derive from my exploration of the periodic system from a fresh perspective in the past few years. I had found that the more electronically quantum-faithful mapping of the periodic table (the Janet Left-Step table from 1929, see the Wiki piece on alternative periodic tables) had numerical milestones that connected it to Pascal's Triangle. In the Janet system all periods of same length come doubled (periods there end with the s2 configuration, so include He). Because the lengths of periods are half or double squares each doubled set, called a dual, has a square number of elements. Atomic number where the duals are completed are identical to every other Pascal Triangle tetrahedral number (4,20,56,120...) each value representing the running sums of squares of even integers. And, interestingly, counting backwards within any Janet period from the s2 position, using only Pascal Triangle TRIANGULAR numbers, you land always on positions where the quantum number ml is equal to 0 (that is, the midpoints of each orbital half-row).

The nuclear shell system, though it motivates its features in a reordered fashion relative to the electronic one, still uses Pascal relations, but DOUBLED. That is, the major trend is for half the (semi)magic numbers to be doubled Pascal tetrahedral numbers, and the other half to be doubled tetrahedral numbers minus doubled triangular numbers. I'm still looking for further mathematically motivated structure in the nucleus. But be aware that others had already discovered that the in the nucleus there was a trend for N/P to approach the Golden Ratio at the limit of stability (already around 1.58 for uranium).

Back in the electronic system there are more Pascal relations. As many of you know besides the diagonals (which by the way allow one to create a nice tetrahedral periodic representation utilizing close packed spheres where period duals are skew rhombi packed in ever larger tetrahedral arrangements, with p,d,f forming successive sleeves around a central s axis) there are also shallow diagonals that cut across the other, and summing the samplings of these latter give the Fibonacci numbers (using a sister Pascal system with 2's down one side and 1's down the other gives both Lucas and Fibonacci numbers as summed samplings across diagonals).

On a whim I mapped Fibonacci numbers to the periodic system AS atomic numbers and found that every one of them up to 89 (the last one before one hits the Bohr FSC) map to LEFTMOST positions within orbital half-rows, with the first singlet and doublet valence electrons. Up to 89 there are no exceptions. Another incredible coincidence, like the generation of primes using the equation discussed in this thread. All the ODD Fibs mapped to singlets (left half orbital) and all the EVEN ones to doublets (right half). Try it yourself! One should never just take anyone's word for it.

Then I tried the Lucas numbers, and found that up to 18 anyway, they mapped similarly to Fibonacci as atomic numbers, but to the RIGHTMOST edge of the orbital half-rows, where one has the LAST singlet or doublet valence electron. What is fascinating here is that for 29 and 47, copper and silver in the same vertical group, and in the same period dual, both are exactly one position left of where they *should* be trendwise, that is d10, yet they both have d10 'anomalous' ground state electronic configurations. There are more than 20 of these known, which muck up the otherwise perfect quantum mapping of the Janet Left-Step table. Copper and silver achieve this by shifting one electron from the filled s2 orbital out of s to the d9 that these elements should have based on their chart position. They end up with s1d10- where both of these are now in perfect register with the Lucas mapping trend, despite the positional mismatch in the table!

The next Lucas number, 76, osmium, makes a different 'fix'- in terms of chemical behavior rather than ground state configuration. By allowing its s2d6 configuration to behave as if it were actually s2p6, it ends up behaving as if it were to a large degree the noble gas xenon- both s and p being filled. And such reinterpretations are not unknown elsewhere in the system- there are many interblock periodic parallels no longer obvious in the currently popular periodic table. The tetrahedral model of the system captures most of them because there are more straight-line axes in 3D than there are in 2D. It will be interesting to see, should we ever synthesize element 123 (in a few years? Some believe 120 will be the end since it ends a Janet dual, and the next terminus will be at 220!

Anyway, I just wanted to lay the groundwork here so you might better understand where I'm coming from. I don't know for sure whether the FSC work dovetails with the periodic system yet. The latter has several areas where predicted properties of atoms don't' work, due to relativistic effects as well as to differential electronic shielding of the orbital types, spin-orbit couplings, etc.

Fitting all the mathematical pieces into place will be an ongoing challenge- I've found all sorts of interconnections that lead back to Fib and Luc numbers just within the electronic system alone I haven't mentioned here, a kind of background meshwork if you will. What does it all mean????

Jess Tauber