*Post by Archimedes Plutonium*77.58.43.158 jan burse

ETH Zurich

Paul Biran, Marc Burger, Patrick Cheridito, Manfred Einsiedler, Paul Embrechts, Giovanni Felder, Alessio Figalli, Norbert Hungerbuhler, Tom Ilmanen, Horst Knorrer, Emmanuel Kowalski, Urs Lang, Rahul Pandharipande, Richard Pink, Tristan Riviere, Dietmar Salamon, Martin Schweizer, Mete Soner, Michael Struwe, Benjamin Sudakov, Alain Sznitman, Josef Teichmann, Wendelin Werner, Thomas Willwacher

World's first valid Prime Number Theorem proof done via Math-Induction//Correcting Math pre-6th ed

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)

*Post by Archimedes Plutonium*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.

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