[QUOTE=Don Blazys;256337]Quoting ewmayer:

I am the worlds most dangerous crank who has clearly disproved
the boundedness of any "crank score", and will [B][I]continue[/I][/B] to do so! :smile:

Don.[/QUOTE]

In fact, this disproof [I]depends[/I] on your continued efforts. I have great faith in you.

[QUOTE=Don Blazys;256345]It is [B][U]you[/U][/B] who posted those numbers.
Therefore it [B][I]must[/I][/B] be you who "got it wrong".[/QUOTE]

My numbers are exactly what I intended. Your interpretation of them is wrong.

[QUOTE=Don Blazys;256345]All I did was show that the "at least" values
of your very own "predictions",
show a [B][I]decline[/I][/B] in the relative error.[/QUOTE]

They do not. They predict that over the next thousandfold increase the relative error will be [i]at least[/i] 0.3 times what it is. The prediction includes the possibility that relative error will increase 10%, 50%, or even 100%.

But I'm not surprised you failed to catch that 'subtlety'.

[QUOTE=Don Blazys;256345]Now you are angry, because your fragile ego has been bruised![/QUOTE]

I'm not angry. Are you?

[QUOTE=Don Blazys;256345](Note that my function produces values of [TEX]\alpha[/TEX]
that fall right in between the above values.)[/QUOTE]

You chose it that way. When the numbers started falling apart around 10^9 or 10^10 (I'm not looking it up in your paper again), you revised the predictions with new terms (including the electron-proton ratio) since the relative error had flattened out on the old prediction. If you take the count high enough -- we literally [i]provided the code for you[/i] -- you'll find that the relative error has flattened out again, and you'll either need to accept that or revise your prediction again.

Paul and I might have continued the search longer if not for that discovery, which shows that your prediction is nothing more than curve-fitting. (Of course if you use standard curve-fitting techniques you get better results, see my earlier post for an example.)

[QUOTE=akruppa;256351]In fact, this disproof [I]depends[/I] on your continued efforts. I have great faith in you.[/QUOTE]

You crack me up.

Quoting CRGreathouse:
[QUOTE]Your interpretation of them is wrong.[/QUOTE]

You "predicted" that the relative errors for
[TEX]\varpi(10^{14})[/TEX], [TEX]\varpi (10^{15}) [/TEX] and [TEX]\varpi(10^{16})[/TEX] are [B][I][U]at least:[/U][/I][/B]

[TEX]\frac{5*18785}{64036270308500-5*18785}=.00000000147[/TEX],

[TEX]\frac{40*18785}{640362730443000-40*18785}=.00000000117[/TEX], and

[TEX]\frac{350*18785}{6403627390950000-350*18785}=.00000000103[/TEX],

Now, I interpret those numbers as [B][I]declining[/I][/B]
and you interpret them as [B][I]being the same.[/I][/B]

However,
[TEX].00000000147 > .00000000117 > .00000000103[/TEX]

Therefore, my interpretation is correct, and
it is [B][I]your[/I][/B] interpretation that is [U][B]wrong[/B][/U].

Quoting CRGreathouse:
[QUOTE]They predict that over the next thousandfold increase
the relative error will be [I]at least[/I] 0.3 times what it is.
The prediction includes the possibility that relative error
will increase 10%, 50%, or even 100%.[/QUOTE]

The fine structure constant is known to only 9 significant digits.
Using it to calculate 16 and 17 digit values of [TEX]\varpi(x)[/TEX] and then
claiming that the relative error "increases" is something only
a "mathematician" of your tiny caliber would do!

Quoting CRGreathouse:
[QUOTE]
You chose it that way.
[/QUOTE]
No, it emerged that way naturally.

Quoting CRGreathouse:
[QUOTE]
When the numbers started falling apart...
[/QUOTE]
The numbers never started "falling apart".

In the function [TEX]B(x)[/TEX], there is only [B][U]one[/U][/B] value of [TEX]\alpha[/TEX] that
results in a relative error that approaches some constant,
and that value is just about [TEX]1/137.035999084[/TEX].

No other value of alpha will do, so it's [B][I]required[/I][/B] and not "chosen".

The relative error that is then approached tells us what the "error term" should be.

Quoting CRGreathouse
[QUOTE]
If you take the count high enough you'll find that the relative error has
flattened out again, and you'll either need to accept that or revise your prediction again.
[/QUOTE]
Like I said, taking the count "high enough" using a value of [TEX]\alpha[/TEX]
that is good to only 9 digits is something only a bird brain would do!

Quoting CRGreathouse[QUOTE]
Paul and I might have continued the search longer if not for that discovery.
[/QUOTE]
You "stopped your search" (or so you say) just before you got to [TEX]\varpi(10^{14})[/TEX]
because you were [B][I]too embarrassed[/I][/B] to post that result!
You just can't admit when you are wrong.

Quoting CRGreathouse:[QUOTE]
Of course if you use standard curve-fitting techniques you get better results.
[/QUOTE]
If that turns out to be the case, then I will call the coefficients the "Blazys constants".

It's all good!

Doesn't that make you happy?

Don.

Quoting akruppa:[QUOTE]
In fact, this disproof [I]depends[/I] on your continued efforts.
I have great faith in you.
[/QUOTE]
Thanks, but truly, I couldn't do it without you! :smile:

Don.

[QUOTE=Don Blazys;256495] [B][I][U]at least:[/U][/I][/B]
[/QUOTE]

Note that you can also change the text color if italics, bold and underline don't provide enough emphasis. The color drop-down menu is next to the font size drop-down.

[QUOTE=Don Blazys;256495]Now, I interpret those numbers as [B][I]declining[/I][/B]
and you interpret them as [B][I]being the same.[/I][/B][/QUOTE]

No.

You claim that I claim that the relative error decreases. In fact I do not make this claim. Instead I give lower but not upper bounds on the error. It's true, there are scenarios in which the relative error decreases and my prediction is correct, but there are also scenarios in which relative error stays the same (in which case my prediction is automatically correct) or increases (likewise).

This is a very simple distinction; even a high school student should understand it.

[QUOTE=Don Blazys;256495]The numbers never started "falling apart".[/QUOTE]

See your polygonal number paper, bottom of page 1 and top of page 2. The relative error in your original prediction [TEX]\mathcal{B}(x)[/TEX] appears to stabilize on a constant value. (You report it to too great precision, but nevermind that.) This is evident around 10^10.

To address that you made a new prediction, [TEX]\mathcal{B}(x)\left(1-\frac{\alpha}{\mu-2e}\right).[/TEX] This prediction similarly seems to stabilize to a constant value. This is evident around 10^14; your changes only bought you a factor of 10,000.

[QUOTE=Don Blazys;256495]Like I said, taking the count "high enough" using a value of [TEX]\alpha[/TEX]
that is good to only 9 digits is something only a bird brain would do![/QUOTE]

Ah! Fantastic. You've now entered realm of Poperian nonfalsifiability, along with such fields as astrology and channeling.

I can give you a hint for escaping, but you won't like it: the square root term. Your coefficient there (similarly defined in terms of alpha and other constants) is off by a great deal, as I showed several pages ago. If you plot the function minus the linear term over the square root you'll see that it's quite stable at a value not matching yours even to three decimal places.

[QUOTE=Don Blazys;256495]You "stopped your search" (or so you say) just before you got to [TEX]\varpi(10^{14})[/TEX]
because you were [B][I]too embarrassed[/I][/B] to post that result![/QUOTE]

Hah! Now you're trying to con me into giving you free computer resources!

I have too many other problems to work on, mostly at the OEIS. (If you look up my recent work you'll see exactly where those cycles went.) Now if you were willing to actually knuckle down and do the work yourself, that would be different. Paul's already done the intellectual work for you, providing you with the software, but that's not enough for you -- you want someone else to provide you, [i]gratis[/i], with the computing power!

No, I went just far enough to see that the error did what we all (all but you!) suspected it would. Once it did it didn't take long for Paul and myself to loose interest. Having done that we both devoted our programming time and CPU cores to other (frankly more interesting) work.

But hey! Have fun with your project regardless. Let us know if you ever manage to calculate up to 10^15, where you'll find the [absolute] error is roughly ten times what it was at 10^14.

[QUOTE=Don Blazys;256496]Quoting akruppa:
Thanks, but truly, I couldn't do it without you![/QUOTE]

I feel compelled to explain this, lest others continue to have fun at your expense. The "crank score" is of course the measure of how crazy someone (in this case someone claiming to do mathematics) is. If the crank score was bounded, it would mean that there's a fixed 'insanity level' beyond which no person may be -- people can be a little crazy, maybe claim to have proved a few theorems that they haven't, but no more. If, on the other hand, a person could have arbitrarily many wacky and false beliefs -- that he has proven, say, the Tijdeman-Zagier conjecture, Fermat's Last Theorem in just a page, the nonexistence of odd perfect numbers and Lucas pseudoprimes, etc. -- then we'd say that the crank score was unbounded.

Quoting akruppa:
[QUOTE]
Note that you can also change the text color if italics,
bold and underline don't provide enough emphasis.
The color drop-down menu is next to the font size drop-down.
[/QUOTE]
Yes, I know. I sometimes use red or blue to put some emphasis
on persons, places or things that are "extraordinarily important",
such as scientists and mathematicians of unparalleled genius.

However, all those "text features" are also designed so that
[B][I]special[/I][/B] emphasis can be given to certain words and phrases
in order to help those who are not particularly gifted in math
better understand it.

I often wish that I didn't have to use those text features for
that purpose, but unfortunately, I do.

Quoting CRGreathouse:
[QUOTE]
It's true, there are scenarios in which the relative error decreases...
[/QUOTE]
[B]Finally[/B], you admit that!

Quoting CRGreathouse:
[QUOTE]
...even a high school student should understand it.
[/QUOTE]
I agree.

Quoting CRGreathouse:
[QUOTE]
The relative error in your original prediction [TEX]B(x)[/TEX]
appears to stabilize on a constant value.
[/QUOTE]
Yes it does, but the [B][I]important[/I][/B] thing to note is that
there is only one [B]unique[/B] value of [TEX]\alpha[/TEX] for which this is true.

Quoting CRGreathouse:
[QUOTE]
This prediction similarly seems to stabilize to a constant value.
This is evident around 10^14.
[/QUOTE]How would you know?
You claim that you never calculated [TEX]\varpi(10^{14})[/TEX].

Quoting CRGreathouse:
[QUOTE]
You've now entered realm of Poperian nonfalsifiability,
along with such fields as astrology and channeling.
[/QUOTE]
The [B]fine structure constant[/B] has been experimentally determined to,
at most, 9 significant digits, and many scientists and physicists think that
this mysterious constant is actually known to only 5 or 6 significant digits.

Moreover, some scientists now believe that the [B]fine structure constant[/B]
[B][I]varies[/I][/B] throughout the universe, and may even have upper and lower bounds
with only the first 3 or 4 digits in common.

These are hard cold and undeniable scientific facts which you can easily
check for yourself. However, this does not mean that you should now
compare scientists and physicists to "astrologers" and "channelers".
After all, it is clearly not [B][I]their[/I][/B] fault that the origin and value of
the [B]fine structure constant[/B] is so poorly understood.

Even Wikipedia gives several different values of the [B]fine structure constant: [/B]
137.035999084(51) (it's value measured at an energy of .511MeV), and
137.036, (it's theoretical value at zero energy), but you [B][I]still[/I][/B] don't seem to
understand that your predictions are based on an [B][I]arbitrarily[/I][/B] chosen constant.

Quoting CRGreathouse:
[QUOTE]
I can give you a hint for escaping, but you won't like it: the square root term.
[/QUOTE]
Of course I like it! After all, who [B][I]wouldn't[/I][/B] enjoy having a couple of
extremely useful and therefore extremely important mathematical
constants named after themselves? In fact, I have several friends
who have given me many [B][I]different [/I][/B]values for the [COLOR=red][B]Blazys[/B][/COLOR] constants,
all of which give exellent fits up to at least [TEX]x=10^{12}[/TEX].

We will see which ones are the best when my coder friend resumes
his calculations and determines ever higher values of [TEX]\varpi(x)[/TEX].

Quoting CRGreathouse:
[QUOTE]
I have too many other problems to work on, mostly at the OEIS.
[/QUOTE]
Then we do have at least one thing in common because the OEIS
references this very counting function that we are discussing.

Quoting CRGreathouse:
[QUOTE]
Having done that we both devoted our programming time and
CPU cores to other (frankly more interesting) work.
[/QUOTE]
Most scientists consider the true value and origin of the [B]fine structure[/B]
[B]constant[/B] to be one of the most important unsolved problems in physics.

By contrast, all you are doing is playing childrens games with numbers.

Quoting CRGreathouse:
[QUOTE]Let us know if you ever manage to calculate up to 10^15,
where you'll find the [absolute] error is roughly ten times what it was at 10^14.[/QUOTE]
Again, present determinations of the [B]fine structure constant[/B]
are only accurate to about 6, or at most, 9 digits, which is
insufficient for estimating 14 digit numbers.

Given values such as [TEX]\varpi(10^{15})[/TEX], we first need to solve for [TEX]\alpha[/TEX],
and determine [B][I]it [/I][/B]to 15 digits or so.
Then and only then can we use alpha to estimate [TEX]\varpi(x)[/TEX].

Quoting CRGreathouse:
[QUOTE]
...lest others continue to have fun at your expense.
[/QUOTE]
This entire thread [B][I]is[/I][/B] all about others having fun,
and I don't mind "footing the bill" because
I'm probably having the most fun of all!

Why... with all the distinguished and esteemed mathemticians
continuously and predictably "sticking their feet in their mouths",
"shooting themselves in the foot" and in general "stepping in it",
while trying in vain to refute my work, this thread has become
"more fun than a barrel of monkeys"!

Quoting CRGreathouse.
[QUOTE]
The "crank score" is of course the measure of how crazy someone
(in this case someone claiming to do mathematics) is.
[/QUOTE]
Well, there are other, even more reliable indicators of how crazy someone is.

For instance, only a very, [B][I]very[/I][/B] crazy person would waste precious time
posting on a thread that contains ideas which are either wrong or boring.
(Notice that I have never posted on any of your threads.) :wink:

Yet you, Silverman and a host of distinguished others continue,
continue and... continue to post on my threads, following me,
like so many lost puppies, to the very depths of this "crank subforum"!

Have you ever stopped to think about what that actually means?
Do you know what that tells everybody? Now, look carefully...
[SIZE=1][/SIZE]
[SIZE=1]It means that either I am right, or that you are very, very crazy!
[SIZE=1][/SIZE]
[/SIZE]Don.

On the one hand I think it's pretty neat that we can make mention of someone and then poof! they show up. On the other hand it's not completely fair to assign a huge crank score here; I imagine if the same had happened and [i]Archimedes Plutonium[/i] had shown up instead, we would all be wishing fondly for threads that are only as cranky as this one.

(If you're not familiar with Mr Plutonium, imagine a guy who could singlehandedly make all of usenet unbearable)

[QUOTE=Don Blazys;256677]We will see which ones are the best when my coder friend resumes
his calculations and determines ever higher values of [TEX]\varpi(x)[/TEX].[/QUOTE]

Is your coder friend running his own (slow) code or the one provided by xilman?