World's first valid Prime Number Theorem proof done via Math-Induction


MATHEMATICAL-INDUCTION PROOF OF PRIME NUMBER THEOREM PNT; first valid proof of PNT


Re: beefing up my Math Induction proof Page84, 12-1, World's first valid Prime Number Theorem proof done via Math-Induction

On Wednesday, April 12, 2017 at 3:37:37 PM UTC-5, Don Redmond wrote:
(snipped)
>
> Firstly, I don't see where the induction argument is. It looks more like you're trying to show that the sum of the first n positive integers is 27 and readjusting at each step. That's not a proof, that's the arithmetic equivalence of doodling.
>
> Secondly, why does the sum of the reciprocals of the powers of 2, from 0 onward equaling 2 seem so strange. After N steps you have 2 - 2^(-N). Even on your grid system this will look like 2 for N large enough. Similarly for the sum of the reciprocals of the triangular numbers we get, after N steps we get 2 - (2/(N + 1), which also look like 2 on your grid system for N large enough. We have nothing to doctor here.
>
> Don

Hi Don, yes I can make that much more clear as to what I was Inducting.

Since no computer on Earth can tell us what the primes counting is at 1*10^604, I thought Math Induction is the only means of establishing the proof.

So I assembled a precision definition of when two series are equal. You need a case 1 equality of 100/4 = 25 is our case 1. Then, jumping ahead to the 10^604 Grid, I need to have an equality there. So, given case 1 and the ending case as equal, and everything in between can be Math Induction doctored. I prove PNT true.

What I Induct, is the entire process of doctoring a Grid.

I have case 1, now I inspect Grid 10^4, and doctor it so we have equality.

Now I take Grid 10^5 and do the same thing.

Then I say given case 1 and given the process of Grid 10^4 then Grid 10^5, that we can doctor the end Grid to form a equality, means that I Math Inducted the entire process to have beginning equality and have ending equality.

Now, you may complain that you can prove every series is equal to another series, but not true. For example, the Riemann Hypothesis with the Euler series, there is no beginning equality, meaning that the Riemann Hypothesis is a fakery to even start. Both the Euler series and Riemann series have no beginning equality, hence, the RH is a nonstarter.

Another example. The Even numbers series 2, 4, 6, 8, 10, .... versus Odd numbers 1, 3, 5, 7, 9, .... Are they equal? Old Math says yes. Precision Math says no.

Give me a starting equality.

The 2 does not equal 1

The second sum 2+4 does not equal 1+3.

Never, along those two Series can you get a Starting Block Equality.

With Prime Number Theorem, you easily have a Starting Equality 100/4 = 25 primes.

Don, question, why are mathematicians ever so derelict in defining equality between two series? Why are they so lazy and ignorant in well defining Series Equality? Even a High School kid recognizes that when you say -- "these two series are equal" recognizes that at some specific term we should have the number. But, no, Old Math says two series are equal because at Infinity puff the magic dragon says they are equal.

So, Don, are you not sick and tired of telling young kids, series A is equal to series B because puff the magic dragon at infinity makes them equal. I sure am sick and tired of hearing puff the magic dragon makes them equal.

AP

Re: beefing up my Math Induction proof Page84, 12-1, World's first valid Prime Number Theorem proof done via Math-Induction

- show quoted text -
Speaking of doctoring up, I need to doctor up the above sentence to read this::

"these two series are equal" recognizes that at some specific term we should have the numbers the same for both series.

When I first learned about Series, in College, I was awestruck by the lack of coordination of series concept, the lack of a definition of when one series equals another series, and what really aggravated me, was that no-one in math seems to understand that if you say series A equals series B, that there has to be a STARTING block equality

1+3+5+7+.....

is never equal to

2+ 4+6+8+... because you can never get a nth term that is equal in both

Now, the Series

2+2+4+6+8+.... can equal 1+3+5+7+.... because we have a starting block equality of the 2nd terms

1+3 = 2+2

And thus, with that Starting Block Equality we can build into the Series a doctored ending equality.

So, Old Math is very very pitifully poor on Series equality and the reason is easy to see-- they have a moron definition of infinity-- no borderline.

AP

Re: beefing up my Math Induction proof Page84, 12-1, World's first valid Prime Number Theorem proof done via Math-Induction

- show quoted text -
Speaking of doctoring up, I need to doctor up the above sentence to read this::

"these two series are equal" recognizes that at some specific term we should have the numbers the same for both series.

When I first learned about Series, in College, I was awestruck by the lack of coordination of series concept, the lack of a definition of when one series equals another series, and what really aggravated me, was that no-one in math seems to understand that if you say series A equals series B, that there has to be a STARTING block equality

1+3+5+7+.....

is never equal to

2+ 4+6+8+... because you can never get a nth term that is equal in both

Now, the Series

2+2+4+6+8+.... can equal 1+3+5+7+.... because we have a starting block equality of the 2nd terms

1+3 = 2+2

And thus, with that Starting Block Equality we can build into the Series a doctored ending equality.

So, Old Math is very very pitifully poor on Series equality and the reason is easy to see-- they have a moron definition of infinity-- no borderline.

AP



In New Math we have Grid Systems:

10 Grid
100 Grid
1000 Grid
10^4 Grid
10^5 Grid
10^6 Grid
etc

Primes, the actual count of Primes in those Grids listed above follow this progression


This is the Grid PROGRESSION

10   4 actual primes, 10/2 = 5 predicted when using the formula of base sqrt10 is exponent 2

10^2         25 actual       25 SPOT ON EXACT with sqrt10 base 100/4 which Here is 4

10^3         168 actual       lower bound 1000/7 = 142,  
upper bound 1000/6 = 166   HERE 6

10^4         1,229  actual   lower bound 10,000/9 = 1111 ,  
upper bound 10,000/8 = 1250   HERE 8

10^5         9,592 actual   lower bound 100,000/11 =  9090   ,  
upper bound 100,000/10 = 10,000      HERE 10

10^6         78,498 actual   lower bound 1,000,000/13 = 76,923 ,   HERE 13
upper bound 1,000,000/12 = 83,333  

10^7          664,579 actual     lower bound 10^7/16 = 625,000   ,  
upper bound 10^7/15 = 666,666    HERE 15

10^8          5,761,455 actual      lower bound 10^8/18 = 5,555,555    
 , upper bound 10^8/17 = 5,882,352  HERE 17

10^9         50,847,534 actual     lower 10^9/20 = 50,000,000   ,     HERE 20
upper 10^9/19 = 52,631,578

10^10        455,052,511 actual   lower 10^10/22 = 454,545,454   ,    HERE 22
 upper 10^10/21 = 476,190,476

10^11        4,118,054,813 actual     lower 10^11/24 = 4,116,666,667   ,    HERE 24
upper 10^11/23 = 4,347,826,087

10^12        37,607,912,018 actual   lower 10^12/26 = 38,461,538,46-     HERE 26
upper 10^12/25 = 40,000,000,000


SERIES representation of Actual Primes rather than Progression representation

4 + (25-4) + (168-25) + (1229-168) + (9592-1229) + . .

Progression of Predicted primes from formula using base sqrt10

10/2 =5, 100/4 =25, 1000/6 =166.666.. , 10000/8 = 1250, . .

Series representation of Predicted Primes using Formula sqrt10 as base with its exponent

5 + (25-5) + (166.666..- 25) + (1250 - 166.666..) + . .


ACTUAL INFINITY BORDERLINE IS 1*10^604 but since that is cumbersome to work with and since no computer has ever calculated the actual-primes from 0 to 1*10^604 we PRETEND infinity border is 10^4 and use that via MATH-INDUCTION to find the Doctored Formula.

MATHEMATICAL INDUCTION PROOFS to tell if two different Series are equal to each other at infinity requires these items:
1) Starting Equality
2) Mid Section of N to N+1 provided by Grid System
3) Ending Equality at the infinity border

Here we pretend 10^4 is the infinity border and look to find a DOCTORED formula of base sqrt10 exponent that delivers and Ending Equality.

10^4 Grid has actual 1229 primes yet predicts 10000/8 = 1250

So we Doctor our Formula of base sqrt10 exponent  to use pretend Grid 10^4 where N is 4 exponent on 10000/8 and subtract actual primes of Grid N-2 (that is 10^2 Grid) and add actual primes of Grid N-3 ( which is 10 Grid). So we have arithmetic wise, we have 1250 - 25 + 4 = 1229.

So we have in the end here, we have a Starting Equality of 100/4 = 25 primes in 100 Grid and we have a Ending Equality of 10000/8 subtract 25 add 4 = 1229.

Now, we apply that SCHEME or Pattern to the true infinity borderline of 1*10^604 and so we have something like this 10^604/1208 subtract Grid N-2 add Grid N-3 or add Grid N-1 add Grid N-2.

We do whatever it takes to Doctor the formula so that it makes the End result equal to actual primes at infinity border.

Now, we have a starting equality 100/4 and a Doctored End Equality. Now we do not worry about applying the Doctored Formula on the Mid Section terms, for we certainly have to apply that doctoring. Just so long as we have the Start and End terms agree with actual Prime Count.

Now, let me show you this method on the Zeta Series of the Riemann Hypothesis and show you why they are never equal for no starting-equality is possible at infinity. Show you on the Harmonic series (Oresme) and why it never diverges. And show you on the reciprocal two series of 2-doubled compared to triangular numbers alleged to converge to 2, why they are not equal at infinity border since it is impossible to doctor them at infinity border for a Ending equality. However, on the reciprocals we can find a different series for each that does converge at infinity border and hence are equal series.

What I am developing here is the first time in math history that we clean up the concept of two different Series equaling each other. This has never been done before for the simple reason that when you have Infinity as a notion, mere notion and opinion as to what infinity is, you cannot have a concept of Series equality at infinity. Only when you have a precise infinity with a borderline, can you have precision Series understanding.


Page85, 12-2, many major proofs
AP's Proof of Riemann Hypothesis, part 1 of 2

PART 1 of 2: LOGICAL FLAWS & DISPROOF OF THE RIEMANN HYPOTHESIS //Correcting Math 5th ed

MAJOR LOGICAL FLAWS & DISPROOF OF THE RIEMANN HYPOTHESIS RH

by Archimedes Plutonium

Introduction:

the connection with PNT and RH Re: Page86, Part 2 of 2: LOGICAL FLAWS & DISPROOF OF THE RIEMANN HYPOTHESIS //Correcting Math 5th ed

Now these first two proofs-- Prime Number Theorem PNT and Riemann Hypothesis RH, follow a pattern, a obvious bold and strong pattern.

The pattern is that Old Math has a crazy quilt definition of Series equality. Since Old Math has no borderline with infinity, they have no good definition of what it means for two series to be equal. And so sloppy are they, that they imagine the Series 1 + 1/2 + 1/3 + 1/4 +, . . . the harmonic series is equal to the Series 1+1+1+.... at infinity, Old Math believes those two Series are equal. That is how crazy Old Math was.

And then, of course, when something like PNT comes or RH comes, that having a crazy notion of what is equality for Series, it is little wonder that the nutters of Old Math have troubles with PNT and RH.

AP

Comparing RH to the Prime Number Theorem PNT, in that proof Mathematical Induction on Series, I had a Starting Equality of 100/sqrt10^4 and using sqrt10 base rather than "e" base I have a Starting Equality of 100/4 = 25, and at infinity I Doctor the Ending Equality.

Now, we compare Riemann Hypothesis with its zetas and we also compare the Harmonic Series. The Zetas and the Harmonic Series are never able to give us a Starting Equality to apply a Math Induction proof of RH. What that means is RH is unprovable. It is a false conjecture.

The reason that Riemann Hypothesis is always a failure, is because the two series of Zetas are never equal to each other term per term, because they lack a Starting Equality to form a proof by Mathematical Induction. The Prime Number Theorem is provable because it has a Starting Equality of 100/sqrt10^4 where we have 100/4 by replacing "e base" with sqrt10, and although the prediction count by using sqrt10 is far more sloppy than other formulas based on "e", there can never be a Starting Equality with "e".

The way to prove PNT or RH is via Math Induction, with a Starting Equality and let the Grid Systems be the Math Induction format of "if N then N+1". and at the Infinity borderline wash away the imperfections for a Ending Equality by Doctoring or by tacking on more terms to Equilabrate each series at infinity. This can be seen in the Proof of the Prime Number Theorem. So long as we have a Starting Equality, we can prove PNT.

Why do the Zetas never equal each other but rather-- asymptotically approach one another? It is because they both have BadFractions such as 1/3 = .333... which the 3 digits go beyond the infinity borderline and never able to be tamed into a Starting Equality. Unlike what are GoodFractions 1/8 = 0.125000...

THE LOGIC of the Mechanics of RH proof: the logic here is that RH should parallel the proof of the PNT. Not that Prime Number Theorem is equivalent to RH, but parallel in proof structure.

I have never compared PNT to RH and how RH should be proved using PNT in parallel concert.

Both PNT and RH have huge problems with Series, but it did not stop PNT from having a proof.

CONVERGENCY THEORY precisely defined:

Convergency theory, and what we have is a less strict form of equality. In math we have equality but also we have convergency to make two concepts, each distinct from one another converge to equality at infinity via Math-Induction on series. Not asymptotic approach of two different series.

Convergency boils down to five items:

i) starting equality terms of two items in comparison
ii) Middle terms close together by a lower and upper bounds factor, a "if N then N+1" which is demonstrated by the Grid System.
iii) the terms near the infinity borderline and are either Doctored of formula or are tacked on terms to Equalibrated both series.
iv) the final terms of the two items in comparison are equal "at infinity"
v) convergency has a very close similarity to how Mathematical Induction works in that a starting equality, a ending equality and we say all the terms in the two items under comparison are "convergent equal" in the Middle section

However, I do see a huge problem in a ** starting equality ** for the Zetas. I managed to find starting equalities in PNT such as the primes from 0 to 100 are 100/4 where the 4 is begot from using (sqrt10)^4 rather than using "e log".

So, I see huge problem in getting a STARTING EQUALITY for the Riemann and Euler Zetas.

So, what I suspect is going to happen is that a proof of PNT is possible because these conditions are able to be satisfied but not satisfied for the zetas of RH.

The RH was invented in a time period of 1800s where infinity was ill-defined and never well-defined and so was the theory of Series. So that the RH was solid Old Math with foggy notion of infinity and a infinity without a border between finite and infinite, so that a Oresme series of 1 + 1/2 + 1/3 + 1/4 + . . in Old Math was considered Divergent, of course because infinity was a screwy notion of "forever without any borders".  In New Math, the borderline is found to be 1*10^604 and so Oresme's Series is a finite number Convergent series as should be all Fractional Term series converge. We see from Grid Systems where we pretend that 100 is infinity border that 100 terms in the Harmonic Series 1+1/2 + 1/3 + 1/4 + . . + 1/100 would converge to approx 5 or thereabouts so that 5% of 100 as infinity border indicates that the larger powers of 10 as infinity border would have a decreasing percent for convergence. So, in Oresme to Euler and Riemann with their ill-defined infinity they would have thought the Harmonic Series diverges when in fact it converges. That effectively puts an end to a proof for RH.

The subject of Logic was never really all that big during the time of both Euler and Riemann and they had errors of logic in their thinking and published work which today's modern day mathematicians have never come face to face with.

So Equality of Series is similar to Mathematical Induction, in that you need a starting case, and then you assume "n" and if "n+1" is true, then the set is equal to the Counting Numbers. For the Zetas, we never have a Starting Equality.

Similar to the Prime Number Theorem of the accounting of the abundance of primes. The formula Li(x) is terribly close to equaling the amount of primes, but it too suffers from never having a starting equality, because the logarithmic function Ln can never give equality. So when I replace Li(x) with sqrt10 as basis, we see that for 100/4 where the 4 is (sqrt(10))^4 gives exactly 25 primes from 1 to 100. So we have a Starting Equality and at the Infinity borderband we can engineer a ending equality and so we have equality of amount of primes with the formula Sqrt10 basis.

Very crude dot picture of 5f6, 94TH
ELECTRON DOT CLOUD of 231Pu


                ::\ ::|:: /::
                 ::\::|::/::
                     _ _
                    (:Y:)
                     - -
                 ::/::|::\::
                ::/ ::|:: \::
One of those dots is the Milky Way galaxy. And each dot represents another galaxy.
            . \ .  . | .   /.
           . . \. . .|. . /. .
              ..\....|.../...
               ::\:::|::/::
---------------      -------------
--------------- (Y) -------------
---------------      --------------
               ::/:::|::\::
              ../....|...\...
           . . /. . .|. . \. .
            . / .  . | .   \ .

 
http://www.iw.net/~a_plutonium/ 
whole entire Universe is just one big atom 
where dots of the electron-dot-cloud are galaxies

I re-opened the old newsgroup PAU of 1990s and there one can read my recent posts without the hassle of spammers, off-topic-misfits, front-page-hogs, stalking mockers, suppression-bullies, and demonizers.     

https://groups.google.com/forum/?hl=en#!forum/plutonium-atom-universe        
Archimedes Plutonium

Now i put the PNT together with Riemann Hypothesis for good reason. Both are dead in the water, until mathematicians well degine what it means for series A to equal series B. This bs of mathematicians saying A equals B at infinity has got to stop. Such utter ignorance as saying 1+ 1/2 + 1/3+,.... equals 1+ 10 + 100+,..... at infinity is total bs not worthy of even an insane asylum.

You see for PNT it starts out great, fantastic y/Ln(y) in base 10 is 100/4 = 25 primes and then there are EXACTLY 25 primes in reality. But, for the rest of PNT and for the Riemann Hypothesis everyone in math gave a bs about when a Series is equal, equal to some quantity or to another series. It was like a thousand year sleep in math to define series properly.

AP

On Friday, April 28, 2017 at 6:55:17 PM UTC-5, Don Redmond wrote:
(snip)

> For the harmonic series 1 + 1/2 + 1/3 + ... +1/n this finite sum is approximately ln(n). Are you going to say that the logarithm is a bounded function? If it isn't then, you'll be able to get beyond your infinity border in no time, numerologically speaking.
>
> Don

Hi Don, well, in the next edition of this textbook-- edition 6th, sometime next year, I should devote some pages to defining Series equality or equivalence.

In your Old Math, Don it was impossible to even start defining Series equality because there was no infinity that was well defined. So in Old Math-- every thing that "went to infinity" could be and was fudged over, like a huge gigantic tax loophole that citizens would love because the person paying taxes on 1+10+100+ 1000 + pays no more than the person paying on 1 + 1/2 + 1/3 + 1/4 +...

And the Series of Old Math is a good reason that so many butts of jokes are leveled on mathematics-- I mean, to screw up such a lovely science as Series and make it a cesspool mess is shameful.

So in the next edition, I need a chapter that does this---------

Start with the 10 Grid, pretend that 10 is the infinity borderline for 1*10^604 is way too cumbersome.

So the 10 Grid is all the Rationals of .1, .2, .3, . . , 9.9, 10, or 100 Rationals in all

Now, every Series or Sequence for that matter can have only 10 terms in it, since 10 integers in 10 Grid, or we can be a little lax and allow 100 terms to cover every Rational in 10 Grid, but let us say strictly on Integers. For if we wanted 100 Terms, we can start out with 100 Grid.

Now every Series or Sequence can have just 10 terms in total, because 10 is the Infinity borderline.

So, now, take the Harmonic Series 1+1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10.

Now, Don, does it diverge-- meaning, is it more than 10? Obviously not

The smallest divergent series is nine terms of 1 with the last being 1.1, such as 1 + 1 +. . +1.1 and it equals exactly 10.1 with .1 over infinity. Now if all ten terms were 1 then it is the largest possible convergent series.

Now the Harmonic Series in Old Math was divergent-- only because mathematicians had their heads screwed on backwards. Harmonic Series is always convergent.

So, we easily define convergent and divergent.

Now to define equality. To be two Series that are equal, their sums must be equal

The series 1+1+...+1 of ten terms equals 10 and the series 0+2+0+2 for ten terms sums to 10 also and these two series are equal.

Now notice that in the 2nd term of the above two series both equal 2, in that 1+1=2 and 0+2 = 2.

So, here we precisely define Series Equality.

So, in the Prime Number Theorem, we have a starting equality of x/Ln(x) in base 10 where 100/4 = 25 primes count exactly. We further show that at infinity, you can engineer a equality of x/Ln(x) and thus by math induction have a proof of PNT.

However, with Riemann Series and Euler Series in Riemann Hypothesis, you never have a starting out equality, for the two series were never equal to each other, and Euler gave a phony fake proof of equality.

And throughout mathematics of unproven conjectures where Series has a major portion of the conjecture, that those series are muddied and murky and the reason for no proof.

AP

So Don is complaining about the logarithm function which to him has always been divergent, only because never a precision infinity was used.

But if we have a infinity borderline and if we define a Logarithmic Series, we see it is not divergent but convergent.

In fact with functions we can say every function of 45 degrees or less is convergent and every function more than 45 degree slope is divergent. The 45 degree angle of the identity function is a border angle itself.

AP

So i am trying to remember of any other famous math proof involving series. RH involved two series. PNT involved the series that is the count-quantity of primes and the series represented by x/ Ln(x)

So is there another famous proof involving series?

AP

Correct me if wrong but is not every Integral in Old Math -- at root-- a Series?? If so then the fundamental theorem of calculus et al are flawed.

AP

So, i have to review again the Fundamental Theorem of Calculus as to how apparent or not apparent is the Series concept and what role it has. You see the Limit concept so much destroys math understanding for that is why the Limit was invented by Cauchy in the first place-- tired of his students asking how a rectangle with width of 0 can have an interior area.

The limiit was not invented for understanding anything, to the contrary, was invented to stamp out understanding. To stamp out the discovery of the infinity borderline so that the infinitesimal has some meat on its bones-- a nonzero width.

AP

Archimedes Plutonium, delusional idiot, wrote:
> The limiit was not invented for understanding anything, to the contrary, was invented to stamp out understanding.

So why everyone but you, baba King and John Gabriel, understand limits
very well?

You'd better consider that the problem is on your side, Mr Pluto.

You also find meat in the limit of a series. Take
the series sn = sum_{k=1}^n 9/10^k, or much simpler

    sn = 1-1/10^n

Then we have:

    s = lim n->oo sn = 1

So 0.999... gives 1. Now where is the meat? Very
simple consider the differences:

    Dsn = |s - sn|

We have very simple:

    Dsn = 1-(1-1/10^n) = 1/10^n

So there is always some meat between s and sn, for
ever, for all n. Always, also beyond 10^604!

He He. Nothing is stamped out. LoL, here
have a Banana Dragon Mr. Pluto:

How to breed BANANA - Dragon Mania Legends
https://www.youtube.com/watch?v=leRqGE6nQ6w

Ah, yes,yes,yes,yes,yes,yes,yes!

It is easy to spot, spot the error of Series in the Fundamental Theorem of Calculus.

I have the Stewart Calculus of Old Math to his alleged FTC proof on page 396 (5th edition). Can you spot the huge and laughable mistake. Would you like me to hold of to give you a chance to spot it?

Well then do not read any further for i explain the laughable mistake below.


Stewart uses the Extreme Value Theorem (page 281) which is related to Mean Value Theorem (page 291), only in MVT we end up with tangent lines slanted to x-axis, but in EVT we end up with tangents not slanted but parallel to x-axis. And a contradiction to the starting out function. In other words the combination of fake limit with fake rectangles of 0 width climaxes in a fake proof. In New Math we do get a true proof of FTC by simply noting every slanted line of the rooftop of a partition if you find the midpoint of partition and cut the small triangle and fill in the gap forming a rectangle-- we do prove FTC, not by EVT but by noting all trapezoids from their midpoint forms a rectangle-- and thus slope is intimately connected to area.

AP