Thanks for the info science_man_88.

Don.

Last Friday, the coder working on this theory with me determined that

[TEX]\varpi(1,200,000,000,000)=768,434,854,386)[/TEX]. My function gives:

[TEX]B(1,200,000,000,000)*\left(1-\frac{\alpha}{\mu-2*e}\right)=768,434,853,414[/TEX] for a difference of [TEX]-972 [/TEX]

and a relative error of [TEX]-.00000000124[/TEX], which is outstanding!

An incredibly accurate prediction!

The determination of [TEX]\varpi(1,300,000,000,000)[/TEX] should be complete later today.

Don.

[QUOTE=Don Blazys;253669]Moreover, he is not the only one who thinks that my proof of BC is both true and correct.
If you look at the "articles and letters" on my website, then you will find a lot more evidence that I'm right!
Even the [B]Journal of the London Mathematical Society gave my proof some support[/B] while declining to
publish it due only to a lack of available journal space, and back then my proof was a handwritten manuscript![/QUOTE]

If the London Mathematical Society thought the proof was correct they would have published it. Major outstanding conjectures in exponential Diophantine theory aren't ignored. But of course they saw that the claimed proof was wrong and declined it.

[QUOTE=Don Blazys;253669]By contrast, there is absolutely [B]no evidence whatsoever[/B] that my proof of BC is wrong.
You calling it "utter rubbish", and kids with names like "Punky Munky" calling me a "crank"
in no way constitutes evidence, and I can be assure you that if a fatal flaw was ever found,
then I would drop my proof like a hot potato! I would [B][U]never[/U][/B] waste my [B][I]precious[/I][/B] time on a lie.[/QUOTE]

I've given specific criticisms on a number of different forums. For example:
[url]http://www.physicsforums.com/showthread.php?t=301139[/url]

After the first half-dozen mistakes I lost interest in pointing them out. Had your proof been sound except for those things I might have kept up interest longer, but the whole proof is fatally flawed because of your bizarre assumption that the integers are closed under root extraction.

[QUOTE=Don Blazys;253669]Thus, the controversy continues, and I think that a [B][I]formal[/I][/B] online debate between
a [B]recognized panel of experts[/B] and [B]myself[/B] is in order. Maybe you can help arrange that![/QUOTE]

I really couldn't -- the experts wouldn't spend their time on a 'proof' like that. If the proof looked correct to me I *might* be able to get certain professors I know to review it on the strength of my recommendation (as a favor), but since it doesn't I wouldn't even be able to convince them on those grounds.

Quoting CRGreathouse:

[QUOTE]
If the London Mathematical Society thought the proof was correct they would have published it.
Major outstanding conjectures in exponential Diophantine theory aren't ignored.
But of course they saw that the claimed proof was wrong and declined it.
[/QUOTE]That's not true.

They declined for the exact same reason they decline to publish 99% of
the papers that are submitted to them, which is lack of journal space.

The London Mathematical Journal found [B][U]no[/U][/B] fatal flaw in my paper.
Quite the contrary, the referee gave my work [B][I]some support ! [/I][/B]
If there was a fatal flaw, then they would simply have pointed it out.
Instead, they recommended that I send it to another [B]good[/B] journal!

That's because it's correct! :smile:

Quoting CRGreathouse:
[QUOTE]I've given specific criticisms on a number of different forums. For example:
[URL]http://www.physicsforums.com/showthread.php?t=301139[/URL]
[/QUOTE]In that rude and childish forum, your single "criticism" about "integrality"
needed only a little clarification and in no way constitutes a "fatal flaw".

I encourage everyone to read that entire thread and see for themselves that
no one who participated in that discussion ever found a "fatal flaw" in my proof.
In fact, everyone who tried made complete and utter fools of themselves.
(And I was being as gentle as I could be!)

Finally, their fragile egos couldn't take anymore and in their frustration,
they locked that thread, thereby capitulating and admitting defeat.

(By the way, if you Google search "Beal's Conjecture Proof", then you will find that
the above thread that you linked to is #3. Clearly, it is very popular because
I cleaned their clocks! In fact, I [B][I]still[/I][/B] get e-mails about it!)

Quoting CRGreathouse:
[QUOTE]
After the first half-dozen mistakes I lost interest in pointing them out.
[/QUOTE]There were no "half dozen mistakes".

The [B][U]truth[/U][/B] is... I made [B][I]no[/I][/B] errors. In fact, on post #28 of that thread,
there is a list of all the errors that were made, and my only "faux pas"
was to not fully explain something that I thought was obvious.

Quoting CRGreathouse:
[QUOTE]
Had your proof been sound except for those things
I might have kept up interest longer,
but the whole proof is fatally flawed because of your
bizarre assumption that the integers are closed under root extraction.
[/QUOTE]I would [B][I]never[/I][/B] claim or assume such a silly notion!

Here's my proof:

[U][COLOR="Navy"]httр://donblazys.com/02.рdf[/COLOR][/U]

please [B][I][U]show[/U][/I][/B] me where I claim that integers are closed under root extraction.

Quoting CRGreathouse:
[QUOTE]
...the experts wouldn't spend their time on a 'proof' like that.
[/QUOTE]That's wrong too.

A lot of experts [B][I]did[/I][/B] spend a lot of their time on it, (years in some cases!)
and found my Proof of Beal's Conjecture to be both true and correct. :smile:

Anyway, this thread is supposed to be about my polygonal number counting function
which is something that number theory desperately needs because
polygonal numbers of order greater than 2 are so incredibly hard to count.
([TEX]\varpi(1,300,000,000,000)[/TEX] should be determined by tomorrow... )

However, if you would like to continue discussing my Proof of Beal's Conjecture,
then please let me know, and I will start a new thread on it.


Don.

[QUOTE=Don Blazys;254162]
Here's my proof:

[U][COLOR=Navy]httр://donblazys.com/02.рdf[/COLOR][/U]
[/QUOTE]

I know I perhaps should not do this, but... I did have a look at the files on you web pages.

For example when you write equation (1) you assume that ([TEX]z = 1[/TEX] or) [TEX]c \ne T[/TEX]. (Otherwise the last equality does not hold (assuming you are using real numbers here).) The conclusion which follows "and division by zero prevents - -" is therefore [B]false[/B].

[QUOTE=Don Blazys;254162]
They declined for the exact same reason they decline to publish 99% of
the papers that are submitted to them, which is lack of journal space.

The London Mathematical Journal found no fatal flaw in my paper.
Quite the contrary, the referee gave my work some support !
If there was a fatal flaw, then they would simply have pointed it out.
Instead, they recommended that I send it to another good journal!
[/QUOTE]

Perhaps you are talking about some other letter than the one in [U][COLOR="Navy"]httр://donblazys.com/letters_and_articles.рdf[/COLOR][/U]?

Assuming the odd case that it is the same letter, then.. The letter reads "- - we felt obliged to reject your paper in favour of more highly recommended contributions." which [B]does not[/B] imply that they did not publish it because of lack of journal space. The letter also [B]does not[/B] say that the LMS found no fatal flaw; you can, however, say they did not report any. Also, they "hoped that you have no difficulty in finding another good journal for the paper" which is [B]not[/B] the same as recommending sending it to another good journal.

(It could well be that you made these conclusion after some extra communications with the LMS; or that you are talking about some other submission.)

About the [I]A Special Polygonal Number Counting Function Involving the Fine Structure Constant and the Proton to Electron Mass Ratio[/I]: Do you have any (heuristic) arguments why the approximation should be correct or is it a result of numerical experiments? Have tried any probabilistic (or other trivial) approximations to obtain bounds?

Quoting rajula:
[QUOTE]I know I perhaps should not do this, but... [/QUOTE]I think that you will be okay, just as long as you are [B]very careful[/B]
and proceed with [B]great caution[/B]. Don't forget, I am not just "[B][I]any[/I][/B] crank",
I am a "[B][I]dangerous crank"[/I][/B] who will "get you" with his "counting function"!:lol:

Quoting rajula:
[QUOTE]
...when you write equation (1) you assume that [TEX]z=1[/TEX] [/QUOTE]Here is my proof:

[U][COLOR="Navy"]httр://donblazys.com/02.рdf[/COLOR][/U]

As anyone can see, equations (1) and (2) assume [TEX]Z,z>2[/TEX].
It is then (and only then) that [TEX]T\not=c[/TEX], which means that substituting [TEX]\frac{c}{c}[/TEX] for [TEX]\frac{T}{T}[/TEX] is impossible,
which is a contradiction, because it [B][I][U]must[/U][/I][/B] be possible to substitute [TEX]\frac{c}{c}=1[/TEX] for [TEX]\frac{T}{T}=1[/TEX].

Equation (3) assumes [TEX]Z=1[/TEX] and equation (4) assumes [TEX]z=2[/TEX] which completes the proof because
both equations (3) and (4) allow [TEX]T=c[/TEX] which in turn allows us to substitute [TEX]\frac{c}{c}[/TEX] for [TEX]\frac{T}{T}[/TEX] which means that
there is no contradiction if and only if [TEX]Z=1[/TEX] and [TEX]z=2[/TEX].

Quoting rajula:
[QUOTE]
(assuming you are using real numbers here)
[/QUOTE]"Real numbers" are [B][I][U]not[/U][/I][/B] assumed because they include "irrationals"
for which the concept of co-primality is meaningless!
As anyone can see, in my proof, each and every variable is
carefully defined as an element in some set of natural numbers.

Quoting rajula:
[QUOTE]
The letter also [B]does not[/B] say that the LMS found no fatal flaw;
you can, however, say they did not report any.
Also, they "hoped that you have no difficulty in finding another good journal for the paper"
which is [B]not[/B] the same as recommending sending it to another good journal.[/QUOTE]You are not being very logical. Now, think about this carefully.

Would the Journal of the London Mathematical Society (which is one of the worlds most prestigious journals)
"hope" that a "fatally flawed" proof would make it's way into another [B][U]good[/U][/B] journal?

Would the Journal of the London Mathematical Society (which is one of the worlds most prestigious journals)
give [B][I][U]some support[/U][/I][/B] to a proof that was "fatally flawed"?

Of course not!

But they [B][I]would[/I][/B] give [B][I][U]some support[/U][/I][/B] to a proof that is both true and correct!

That's why they gave [B][I]my[/I][/B] proof [B][I][U]some support![/U][/I][/B]

Doesn't that make you happy? :smile:

Quoting rajula:
[QUOTE] It could well be that you made these conclusion after some extra communications with the LMS.
[/QUOTE]I phoned them "long distance". They told me that my proof was correct,
but that it was not a "priority" because Wiles already proved "Fermat's Last Theorem".
I then reminded them that my paper proves the "general case",
and they said they would "consider it".

That was a dozen years ago.

I then decided that it would be a [B][I]lot[/I][/B] easier to publish it in an online journal for amateurs,
where it can [B][I]easily[/I][/B] be viewed and "refereed" by the [B][I]entire math community[/I][/B]
rather than just a few subscribers to the Journal of the London Mathematical Society!

As it turns out, that was the right decision because my proof is now (and has been for quite a while)
[B][I]consistently[/I][/B] in the [B]top five[/B] when you Google search "Beal's Conjecture Proof".

Quoting rajula:
[QUOTE]
About the [I]A Special Polygonal Number Counting Function Involving [/I]
[I]the Fine Structure Constant and the Proton to Electron Mass Ratio[/I]:
Do you have any (heuristic) arguments why the approximation should be correct
or is it a result of numerical experiments? Have tried any probabilistic (or other trivial)
approximations to obtain bounds?
[/QUOTE]It's logically derived. The constants [TEX]\alpha[/TEX] and [TEX]\mu[/TEX] emerged [B][I]naturally[/I][/B] and came as a complete surprise.
I will present those details at a later date. Right now it's more important to get higher counts of [TEX]\varpi(x)[/TEX].
By the way, [TEX]\varpi(1,300,000,000,000)[/TEX] was just determined to be [TEX]832,471,110,338[/TEX].
My "approximation function" predicted [TEX]832,471,109,826[/TEX] which is off by only [TEX]-512[/TEX].
Pretty amazing, don't you think?

Don

[QUOTE=Don Blazys;254283]
As it turns out, that was the right decision because my proof is now (and has been for quite a while)
[B][I]consistently[/I][/B] in the [B]top five[/B] when you Google search "Beal's Conjecture Proof".
[/QUOTE]

okay and for the search nephrotic syndrome + site:webs.com my site nephroticsyndrome.webs.com is first ( this doesn't make it accurate( though I did learn some from my doctor)) just means google thinks it fits the search best. it could be that yours is the most read ( still doesn't prove accuracy).

[QUOTE=Don Blazys;254283]
I think that you will be okay, just as long as you are [B]very careful[/B]
and proceed with [B]great caution[/B]. Don't forget, I am not just "[B][I]any[/I][/B] crank",
I am a "[B][I]dangerous crank"[/I][/B] who will "get you" with his "counting function"!:lol:
[/QUOTE]
I meant that I perhaps should not continue off-topic. The topic of the thread was [I]A Special Polygonal Number Counting Function[/I].

[QUOTE=Don Blazys;254283]
- - which means that substituting [TEX]\frac{c}{c}[/TEX] for [TEX]\frac{T}{T}[/TEX] is impossible,
which is a contradiction, because it [B][I][U]must[/U][/I][/B] be possible to substitute [TEX]\frac{c}{c}=1[/TEX] for [TEX]\frac{T}{T}=1[/TEX].
[/QUOTE]
Substituting [TEX]\frac{c}{c}[/TEX] for [TEX]\frac{T}{T}[/TEX] is possible, but substituting [TEX]c[/TEX] for [TEX]T[/TEX] is not. These two substitutions should not be mixed up.

[QUOTE=Don Blazys;254283]
"Real numbers" are [B][I][U]not[/U][/I][/B] assumed because they include "irrationals"
for which the concept of co-primality is meaningless!
As anyone can see, in my proof, each and every variable is
carefully defined as an element in some set of natural numbers.
[/QUOTE]
Natural numbers are also real numbers.

[QUOTE=Don Blazys;254283]
Would the Journal of the London Mathematical Society (which is one of the worlds most prestigious journals)
"hope" that a "fatally flawed" proof would make it's way into another [B][U]good[/U][/B] journal?
[/QUOTE]
They probably would not. But.. rejection-letters (I have received plenty of those :smile:) are usually written with careful and positive tone. They usually praise the manuscript or encourage to work on it or to submit it elsewhere regardless of the content.

[QUOTE=Don Blazys;254283]
Would the Journal of the London Mathematical Society (which is one of the worlds most prestigious journals)
give [B][I][U]some support[/U][/I][/B] to a proof that was "fatally flawed"?
[/QUOTE]
I thought it was the referee who did that? I know that the Journal of the LMS is one of the better journals. I also have some work which is published in it. To my knowledge they follow the usual procedure of refereeing. First the referee or editors decide if the work might be worth considering for publication. If it is, then a referee will read it, comment on it and recommend it to be rejected/accepted/revised. If your manuscript gets this far you will most likely get a referee's report listing corrections and comments.

[QUOTE=Don Blazys;254283]
Doesn't that make you happy? :smile:
[/QUOTE]
It makes me happy that the journals give nice and encouraging responses. I just wanted to point out that it is better to cite the responses rather than the uncertain conclusions. (After one has received huge amounts of letters accepting and rejecting manuscripts, then there is a possibility that one can make accurate conclusions for oneself. I, personally, am not able to make such conclusions and even if I were, I would never consider making those conclusions publicly!)

[QUOTE=Don Blazys;254283]
- - As it turns out, that was the right decision because my proof is now (and has been for quite a while)
[B][I]consistently[/I][/B] in the [B]top five[/B] when you Google search "Beal's Conjecture Proof".
[/QUOTE]
It is true that your writings are easy to find with Google. And if that was your goal, then you have done well in that regard.

[QUOTE=Don Blazys;254283]
It's logically derived. The constants [TEX]\alpha[/TEX] and [TEX]\mu[/TEX] emerged [B][I]naturally[/I][/B] and came as a complete surprise.
I will present those details at a later date.
[/QUOTE]
I look forward seeing the details.

[QUOTE=Don Blazys;254283]
Right now it's more important to get higher counts of [TEX]\varpi(x)[/TEX].
By the way, [TEX]\varpi(1,300,000,000,000)[/TEX] was just determined to be [TEX]832,471,110,338[/TEX].
My "approximation function" predicted [TEX]832,471,109,826[/TEX] which is off by only [TEX]-512[/TEX].
[/QUOTE]
I do not understand why the higher counts are more important. For me they have only little importance.

[QUOTE=Don Blazys;254283]
Pretty amazing, don't you think?
[/QUOTE]
I would have to analyze the behavior of [TEX]\varpi(x)[/TEX] before I could say if the approximation is amazing or not.

[QUOTE=Don Blazys;254283]substituting [TEX]\frac{c}{c}[/TEX] for [TEX]\frac{T}{T}[/TEX] is impossible[/QUOTE]

What does this even mean? The equations cited, (1) and (2), do not even have [TEX]\frac cc[/TEX] or [TEX]\frac TT[/TEX] in them.

To rajula,

Quoting rajula:
[QUOTE]
Substituting [TEX]\frac{c}{c}[/TEX] for [TEX]\frac{T}{T}[/TEX] is possible, but substituting [TEX]c[/TEX] for [TEX]T[/TEX] is not.
These two substitutions should not be mixed up.[/QUOTE]

You are wrong. Here's why...

Given [TEX]\frac{T}{T}[/TEX] and substituting [TEX]c[/TEX] for [TEX]T[/TEX] results in [TEX]\frac{c}{c}[/TEX].

Given [TEX]\frac{T}{T}[/TEX] and substituting [TEX]\frac{c}{c}[/TEX] for [TEX]\frac{T}{T}[/TEX] [B][I][U]also[/U][/I][/B] results in [TEX]\frac{c}{c}[/TEX].

So clearly, the two substitutions are absolutely equivalent!

Thus, if we can't substitute [TEX]c[/TEX] for [TEX]T[/TEX], then neither can we substitute [TEX]\frac{c}{c}[/TEX] for [TEX]\frac{T}{T}[/TEX].

If we substitute [TEX]c[/TEX] for [TEX]T[/TEX] (or [TEX]\frac{c}{c}[/TEX] for [TEX]\frac{T}{T}[/TEX]) on the left side only,
then we can no longer [B][I]derive[/I][/B] the terms involving logarithms!

Quoting rajula:
[QUOTE]
It is true that your writings are easy to find with Google.
And if that was your goal, then you have done well in that regard.
[/QUOTE]
Thank you!

Journals are dead.

They are quickly going the way of the horse and buggy and
exept for a few elitist snobs, nobody reads them anymore.

Most people [B][I]now[/I][/B] get their information by searching the internet,
and when they search for "Beal's Conjecture Proof", it is [B][I]my[/I][/B] proof,
which is [B][I]obviously[/I][/B] both true and correct, that they will find!

Quoting rajula:
[QUOTE]
I do not understand why the higher counts are more important.
For me they have only little importance. [/QUOTE]

Given sufficiently high counts of [TEX]\varpi(x)[/TEX], we can solve for [TEX]\alpha[/TEX] and find out if
it matches the most accurately determined value of the Fine Structure Constant.

You see, [B]polygonal numbers[/B] are every bit as fundamental and important as [B]prime numbers[/B].
That fact alone makes this the most important counting function this side of [TEX]Li(x)[/TEX].
However, if it turns out that [TEX]\alpha[/TEX] in this function precisely matches
the most accurately measured value of the fine structure constant,
then my counting function will be of that much greater importance to mankind.

Doesn't that make you happy? :smile:

Don.

To CRGreathouse:

Quoting CRGreathouse:
[QUOTE]
The equations cited, (1) and (2), do not even have [TEX]\frac{c}{c}[/TEX] or [TEX]\frac{T}{T}[/TEX] in them.
[/QUOTE]

The equations cited, (1) and (2), [B][I]must[/I][/B] have [TEX]\frac{T}{T}[/TEX] in them.

Otherwise, the terms involving logarithms could not have been logically derived.

So please keep looking as hard as you can!

I'm sure that you [B][I]will[/I][/B] be able to find those cancelled [TEX]T[/TEX]'s. :smile:

Don