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1-HISTORY of AP's four proofs of Fermat's Last Theorem//New Proofs of Fermat's Last Theorem, (1)pure algebra, (2) pure geometry KEY-- 2ab,b^2-a^2,b^2+a^2

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  1-HISTORY of AP's four proofs of Fermat's Last Theorem//New Proofs of  Fermat's Last Theorem, (1)pure algebra, (2) pure geometry KEY-- 2ab,b^2-a^2,b^2+a^2 Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-09-29 01:39 -0700
    Proving the WHY of FLT not just the "What is true" Re: 1-HISTORY of AP's four proofs of Fermat's Last Theorem//New Proofs of  Fermat's Last Theorem, (1)pure algebra, (2) pure geometry KEY-- 2ab,b^2-a^2,b^2+a^2 Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-09-29 16:25 -0700
      Re: Archimedes "Total Failure" Plutonium flunked the Math Test of a lifetime-generation test moroney@world.std.spaamtrap.com (Michael Moroney) - 2018-09-30 02:22 +0000
        Re: Archimedes "Total Failure" Plutonium flunked the Math Test of a lifetime-generation test me <dvhaus.dresden@gmail.com> - 2018-09-30 05:14 -0700

#424501 — 1-HISTORY of AP's four proofs of Fermat's Last Theorem//New Proofs of Fermat's Last Theorem, (1)pure algebra, (2) pure geometry KEY-- 2ab,b^2-a^2,b^2+a^2

1-Newsgroups: sci.math
Date: Fri, 28 Sep 2018 22:59:39 -0700 (PDT)

Subject: HISTORY of AP's four proofs of Fermat's Last Theorem//New Proofs of
 Fermat's Last Theorem, (1)pure algebra, (2) pure geometry KEY-- 2ab,b^2-a^2,b^2+a^2
From: Archimedes Plutonium <plutonium....@gmail.com>
Injection-Date: Sat, 29 Sep 2018 05:59:40 +0000


HISTORY of AP's four proofs of Fermat's Last Theorem//New Proofs of Fermat's Last Theorem, (1)pure algebra, (2) pure geometry KEY-- 2ab,b^2-a^2,b^2+a^2

Alright, I am writing up the history of Archimedes Plutonium's four proofs of Fermat's Last Theorem. I am finished with all four, having spent from 1991 through to September 2018 in perfecting these four proofs. I decided I will include these proofs in my Teaching True Mathematics textbook for juniors in college. 

Alright, let us talk about what is Fermat's Last Theorem FLT. And basically it is about Whole Numbers A,B,C and the equation A^n + B^n = C^n where n is a exponent of whole numbers 3 or larger. And what FLT says is there are no whole number solutions for A,B,C. 

Now, beware, there are plenty of solutions for the equation A^2 + B^2 = C^2 for that is the Pythagorean Theorem. But just increase the exponent to 3 or higher, all of a sudden no more solutions of whole numbers for A,B,C. Now a caution to the students. There are always solutions to the FLT equation, but those solutions carry not whole numbers but things like the cube root of 45 or the cube root of 80. 

So, to the students, FLT is about an equation of exp3 or higher wanting a solution set that is only in whole numbers. And FLT says there are no solution sets like that. 

So, students, do not go barging into FLT with a mindset there are no solutions at all, for there are loads and loads of solutions, only FLT wants a special kind of solution, a whole number solution set. For example in exp2 there is the solution set of 3,4,5 because 3^2 + 4^2 = 5^2. 

HISTORY of AP's four proofs of Fermat's Last Theorem//New Proofs of Fermat's Last Theorem, (1)pure algebra, (2) pure geometry KEY-- 2ab,b^2-a^2,b^2+a^2 

Alright, finally two new proofs of Fermat's Last Theorem. FLT should never have been a tough proof, but it certainly has demonstrated that, as measured by the time it has taken of those pursuing it. Certainly, it has taken up so much of my time through the years. But, as consolation, I have ended up with 4 proofs of FLT, whereas, many of my other proofs such as Collatz, or Riemann Hypothesis, I stopped looking for proofs once I arrived at one proof. 

My whole entire journey through FLT started in 1991 where I stated that the unique feature of the number 2 and 4 where it acts as both multiply and add all packaged into one where 2+2 = 2x2 = 4 = 2^2 this unique feature allows 2 and 4 to be a building block of Pythagorean Triples, and because no such building block exists in exp3 or higher, they cannot have triples as solutions. 

Then in 2014, I changed pace, because I wanted to prove a Generalized FLT, why solutions exist for 
2^3 + 2^3 = 2^4 with prime divisor 2 
3^3 + 6^3 = 3^5 with prime divisor 3 
7^3 + 7^4 = 14^3 with prime divisor 7 
and that all such solutions have a prime divisor, which, when proven true-- all must have a prime divisor to be a solution set, will conquer FLT as a corollary. 

So in 2014 I had proven Generalized FLT by the concept of Condensed Rectangles, with a Common Shared Side of two rectangles being the divisor. You cannot make a new third rectangle if you do not have a common shared side to stack two given rectangles. And thus, out pops FLT quick and easy. 

Fast forward to 2018, and I thought of a trick, where much of mathematics is a trick involved. And I noticed that if I substitute into a exp3 FLT or higher, substitute into A^3 + B^3 = C^3, that I can substitute all the solution sets of exp2 and easily help understand why exp3 and higher have no Whole Number solutions. 

Everyone up to that moment in time shrugged off the solution sets of exp2 and never dared use them to prove FLT. That was a breakthrough pioneering moment in FLT proving. Of course in 2014, I gave a full proof of FLT via Generalized, but here, in 2018, I was going for a direct proof, not a corollary, but a direct proof by what I dubbed as Pure Algebra-- meaning-- no geometry references and a Pure Geometry proof-- meaning-- refering to cubes and cubes cut into two rectangular boxes. These are only names of pure algebra and pure geometry, for the proof mechanism in both come down to being the very same proof mechanism-- an argument of algebra in the end. 

And that argument took me at least two months to arrive at its final moment. 

PROOF MECHANISM of both Pure Algebra and Pure Geometry proofs of FLT: Since I substitute solution sets from Exponent2 into any arbitraray given number of exp3, exp4, exp5, etc. the proof mechanism ends up asking why is it that 2ab,b^2-a^2,b^2+a^2 all three numbers are never Whole Numbers. Essentially, you answer that question and you will have proven FLT. 

Now in exp2, we always have nothing but, yes, nothing but Whole Number solutions to 2ab,b^2-a^2,b^2+a^2 

chose a=1,b=1 and you end up with 2,0,2, chose a=1,b=2 you end up with 4,3,5. 

So, the question or problem of being Whole Numbers is never in doubt with exp2, never, because of how it is constructed solution of 2ab, b^2-a^2, b^2+a^2. Plug into that construction two whole numbers of a,b, well, with out a doubt, it will spit out three whole number solutions. 

But now, go to exp3 or higher, can the machine of construction 2ab,b^2-a^2,b^2+a^2, spit out three whole numbers as solutions. That is all that FLT is about, we know we can have solutions, but are they all three Whole Number solutions. 

So I pick any arbitrary number in exp3, or exp4 , etc and for demonstration purposes I picked 5^3 in exp3 

Is there a whole number for A,B in A^3 + B^3 = 5^3 = 125 

I substitute into that sample a solution set from exp2, in fact all solution sets in exp2 can be wrangled into a substitute , I need not look around for 3,4,5, but could have chosen say 9,40,41 with a k multiplier of 5/41. In fact, we substitute every single solution set of Primitive Pythagorean Theorem into our arbitrary picked number. Every single PPT is substituted. 

SUBSTITUTION: 

Where A^3 + B^3 = 125 = 5^3 picked arbitrarily 

Insertion of a known PPT 3,4,5 

C*C*k + C*C*j = C*C*C 

replacing the C=5, k = 1.8, j= 3.2 we have 

25*1.8 + 25*3.2 = 125 

Here we have A^3 = 45 and B^3 = 80, thus A = 3.556.... and B = 4.308.... 

Alright, well good so far, but we have not stated the Proof Mechanism. 

Why is it that every substitution for every exponent 3 and above, are they coming out as Fractional (what Old Math called irrational?) 

That was the last and final piece I needed to explain. Why is the A, B always coming out as irrationals for exp3 and higher? 

Well, think of it for a moment. You have picked arbitrarily your C value as a Whole Number in exp3 as 5^3 = 125. Now in the PPT, the construction formulas are 2ab,b^2-a^2,b^2+a^2 

And there is a powerful Algebra principle at work here, and so often is it used. The principle is that if A,B,C are solutions to A^m + B^m = C^m where m is any number greater than 1, then so is kA, kB, kC solutions in that formula. 

Notice in these formulas, you are going to always have two odd numbers and only one even number. The even number is always 2ab.  That means both b^2-a^2, b^2+a^2 are always two odd numbers. 
  
So, the proof mechanism, as in our sample problem of 125. Can you have two whole numbers be the cube root of b^2-a^2 , simultaneously the cube root of 2ab. 

That is tantamount to asking can the solutions of cuberoot[(a+b)(a+b)] be whole numbers or 4throot[(a+b)(a+b)] be whole numbers, 5th root..... etc etc. 

For example, 2+3=5 , and 5*5= 25, and (2+3)(2+3) = 4 + 12+ 9 = 25. 

Now, try doing cuberoot of the even number 2ab along with the cube root of the odd number b^2-a^2, and the only whole number to survive that challenge is the number 0. Same goes for exp4 and up. 

The Proof Mechanism of FLT, is the fact that 2ab, b^2-a^2, b^2+a^2 can only deliver Whole Number solution sets in exp2 because they are exp2 built, and since all other exponents of A,B,C of 3 or higher cannot deliver a whole number to both 2ab with b^2-a^2. 

A major algebra math lesson to learn here is that of solving a list of formulas simultaneously 

So we have six formulas: 

(1) 2ab 
(2) b^2-a^2 
(3) b^2 + a^2 
(4) A^3 
(5) B^3 
(6) C^3 

And asked to solve for Whole Number solutions, of the a,b,A,B,C, and so in exponent3 or higher, the only whole numbers that can solve all six formulas is the whole number 0 and no others, but 0,0,0 is not a Triple solution. 

QED 


Now in the next several days I will add more and more 

Very crude dot picture of 5f6, 94TH 
ELECTRON=muon 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.      

Read my recent posts in peace and quiet. 

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

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#424540 — Proving the WHY of FLT not just the "What is true" Re: 1-HISTORY of AP's four proofs of Fermat's Last Theorem//New Proofs of Fermat's Last Theorem, (1)pure algebra, (2) pure geometry KEY-- 2ab,b^2-a^2,b^2+a^2

Alright, I just finished up on 4 proofs of Fermat's Last Theorem, and the one that is the most powerful is the one I started with in around 1991-1993. It is extremely powerful and strange and different from all other math proofs in mathematics, for it answers not only the Proof Statement but it goes beyond the proof statement and answers the WHY, why why of FLT. Most proofs answer the "What is there that is true". But in this proof of FLT, it answers the Why FLT must be true.

And in 1993 I wrote the proof as FLT is true because in exp2 there exists a number where its addition is the same as its multiplication. It is not 1 for 1+1 does not equal 1*1. It is the number 2 with 4 in that 2+2= 2*2 = 2^2 = 4. The 2 and 4 are so unique to mathematics that they allow A^2 + B^2 = C^2 so many cases of triples that satisfy that equation. All because 2 and 4 are building blocks, for the A^2 + B^2 is add, and the C^2 is multiply.

If mathematics (this is the WHY) if mathematics had no number to satisfy N+ N = N*N = N^N = M, then we would not have a Pythagorean theorem and we would not have solutions to any A^N + B^N = C^N for any N as whole number 1 or larger.

So, proof of FLT under that knowledge of a special two numbers in mathematics 2 and 4. For exp3, no number N exists where N+ N +N = N*N*N  = M, the closest any number in math comes to that is 2+2+2 = 2*2*2 = either 6 or 8, off by 2. So the reason no solutions in exp3 is no building block. And the same goes for exp4 or higher.

This is a new, very new class of Proving in Mathematics, for we prove not only the What is true, but we prove Why it is true.

Now in Old Math, they coined a phrase of thought for FLT, by saying FLT is the idea that no two cubes in exp3 can add up to being a new cube in exp3. That is sort of half true, but has mislead almost everyone who attempted to prove FLT. For the error in that thinking is that every given Cube of C^3 has two other cubes when added together equals C^3, with the understanding those two cubes are not whole number cubes.

The gist of FLT is not can you add two cubes to make a new third cube, but rather the entire gist is can you always do it with just Whole Numbers.

So, what I am introducing here, a first in the history of mathematics, is a new wave of how to prove a mathematics problem-- of proof of the WHY, not just a mechanical proof of "what is true" but rather extending further into the math question by a proof of WHY it is true.

So, my first proof of FLT is the most powerful proof that ever there will be of FLT, because it answers the question why is exp2 filled with solution sets, yet all those stop at exp3 and none exists in exp3 or higher.

My proof of FLT as a corollary of Generalized FLT 2014, makes no reference to why on Earth does exp3 and higher stop delivering solution sets. That proof is a pure mechanical "What is true". Same goes for my Pure Algebra FLT proof 2018, and Pure Geometry FLT proof 2018, which both are mechanical proofs that only tell "What is true" and goes through a step by step logic that a few math professors can understand (mind you only a few math professors because even they to this day believe a ellipse is a conic section when you nest a cone inside a cylinder and make that slanted cut, the cylinder delivers the ellipse while the cone delivers a oval-- yet not a single professor of mathematics has the honesty and integrity to admit the ellipse is never a conic, not a single one as of 29 SEP2018, yet AP educated the world on this back in 2016.

So this is a brand new PROOF TECHNIQUE in Math history where the Proof not only proves the What is True, but proves WHY it is true.

And in 1993, Roland Dreier pointed out a main feature of this proof method-- explaining the WHY of FLT, not just the What is True FLT.

From: dreier@durban.berkeley.edu  (Roland Dreier) 
Newsgroups: sci.math 
Subject: Re: 1 page proof of FLT 
Date: 18 Aug 93 14:55:02 
Organization: U.C. Berkeley Math. Department. 
Lines: 42 
Message-ID: (DREIER.93Aug18145502@durban.berkeley.edu> 
References: (CBxp0H.817@dartvax.dartmouth.edu> 
(24s7de$cv4@outage.efi.com> 
(CByoqr.7Aw@dartvax.dartmouth.edu> 
(1993Aug18.173020.10123@Princeton.EDU> 

In article (1993Aug18.173020.10123@Princeton.EDU> 
kinyan@fine.princeton.edu (Kin Chung) writes: 
In article (CByoqr.7Aw@dartvax.dartmouth.edu> 
Ludwig.Plutonium@dartmouth.edu (Ludwig Plutonium) writes: 
LP     Hardy in Math..Apology said words to the effect that the 
LP  understanding of any math proof is like pointing out a peak in the 
LP  fog of a mtn range and you can only point so long and do other 
LP  helps and hope the other person will see it and say Oh yes now I 
LP  see it. But you can not exchange eyeballs. Again I repeat the 
LP  arithmetic equivalent of FLT is that for exp2 there exists a 
LP  number equal under add & multiply i.e. 2+2=2x2=4. Immediately a 
LP  smallest P triple is constructible for exp2 i.e. (3,4,5>. But no 
LP  number exists like 2 for exp3 or higher in order to construct P- 
LP  triples for these higher exp. I am very sorry that I cannot make it 
LP  any clearer than that. Time to take a break and reread Hardy Math 
LP  Apology. 

KC   You also say that a smallest P-triple is constructible for exp2   
KC   immediately from the existence of a number N such that 
KC   N+N=NxN, namely N=2. How do you construct a P-triple given N 
KC   with this property? Please note that I am not asking how you do 
KC    it for exp3, but for exp2. 

Before I continue, let me say that this post does not in any way constitute 
an endorsement of LvP's "proof"; what I am about to explain does not 
extend to exponent 3 in the least. However, things are rather easy for 
exponent two.  (Not to be critical, but you really could have figured this 
out yourself :-) 

So suppose we have an N with 2xN=N+N=NxN. Set a=N+1, b=N+N=NxN. 
Then we get 
                     a^2 = (N+1)^2 = N^2+2xN+1 = 2xN^2+1 
also 
                     b^2 = (N+N)^2 = 4xN^2. 
So 
                     a^2+b^2 = 6xN^2+1. 
Now set c=2xN+1.  Then 
                     c^2 = (2xN+1)^2 = 4xN^2 + 4xN + 1 = 4xN^2 + 2xN^2 + 1       
                      = 6xN^2+1. 
So magically a^2+b^2=c^2, just as desired! ! 

If you can figure out how to do that for exponent 3, make yourself famous. 

Roland 
-- 
Roland "Mr. Excitement" Dreier                 dreier@math.berkeley.edu 

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#424556 — Re: Archimedes "Total Failure" Plutonium flunked the Math Test of a lifetime-generation test

Math Failure Archimedes Plutonium <plutonium.archimedes@gmail.com> fails:

> Alright, I just finished up on 4 proofs of Fermat's Last Theorem

Your homework assignment from Dan is WAY overdue! Here it is again.

> Use any aids. Answer in the space provided.
>
> 1. What is the sine of 45 degrees to 3 decimal places?    ____________
>  
> 2. True or false:  10^604 = 0    ____________
>  
> 3. If A is true and B is false, then A AND B is ____________ (true or false).
>
> 4. If A is true and B is true, then A OR B is ____________ (true or false).

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#424591 — Re: Archimedes "Total Failure" Plutonium flunked the Math Test of a lifetime-generation test

Am Sonntag, 30. September 2018 04:22:54 UTC+2 schrieb Michael Moroney:
> Math Failure Archimedes Plutonium <plutonium.archimedes@gmail.com> fails:
> 
> > Alright, I just finished up on 4 proofs of Fermat's Last Theorem
> 
> Your homework assignment from Dan is WAY overdue! Here it is again.
> >
> > Use any aids. Answer in the space provided.
> >
> > 1. What is the sine of 45 degrees to 3 decimal places?    ____________
> >  
> > 2. True or false:  10^604 = 0    ____________
> >  
> > 3. If A is true and B is false, then A AND B is ____________ (true or false).
> >
> > 4. If A is true and B is true, then A OR B is ____________ (true or false).

If I may add the following:

5. If A is true or B is true, then A OR B is ____________ (true or false).

6. True or false: The ellipse is a conic section.    ____________

(In addition please deliver either proofs or refutations for your claims.)

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