### Monday, May 17, 1999

## FLT proof, rest of the story

A week or so ago I put up a post saying that mathematicians on sci.math were denying the reality of my proof of Fermat's "last" theorem. I didn't go into many details but put out a website. And, I didn't say why the readers of sci.math are justified in totally ignoring what I have.

In case anyone cares, here's the rest of the story. If you decide to wade through what follows (it's not really long), I'll tell you why my proof is correct, why most of you can evaluate it without much difficulty, and possibly why it took someone with a physics background to solve it simply.

I've been looking for a simple proof of FLT since the one given by Wiles' because intuition told me if the thing was true then there was a relatively simple reason. I figured I wouldn't need much more math than I already knew and that I didn't want to go over old (and failed trails) that had been produced in the over 350 years mathematicians have been looking at the problem. But, I knew I would need some help.

It's not the easiest thing to get help or attention from mathematicians on the subject. Seems "amateurs" have been coming at them with proofs for just about all of the 350 years the problem has been around. In fact, some of them can get downright testy about it. So I've been creative. Over the past three years I've made a very public search on sci.math with a decent amount of help. I've also often claimed success only to have to recant later when error(s) were found.

So, the readers of sci.math (and possibly now the readers of sci.physics) have good reason to doubt me other than because of the very strong belief that something simple just *couldn't* have been missed for all of this time.

Now, I get to tell you why my proof is true. On my first webpage I introduce the problem and show why I work on it as x^p + y^p = z^p, where p is an odd prime, and I assume integer solutions x,y,z so that I can prove the theorem by contradiction. On my proof page (the third one), I rewrite this in terms of u,y,k and u,y,f instead of x,y,z (pretty basic at this point), where u = fk, and u and y are integers if x,y and z are.

The neat thing is that the polynomials created by this rewrite have roots for f and k that are limited in a special way. I can easily prove that each set has exactly two real roots with the rest imaginary (complex).

Since I can relate each root from the polynomial using u = fk, I know that the sum of the complex roots for one of the polynomials is just a multiple of the sum of the complex roots of the other.

That's the astounding thing that makes everything else possible, because using that fact, I can relate the first coefficient of one polynomial with the first of the other, since it just equals the sum of the roots. So, I relate the moduli of the sums relative to a number I haven't mentioned yet, which I call "q".

And the result of all this effort is just the relation, z^2 = xy(mod q).

That's as far as I go here because I don't want a really long post. Suffice it to say that using the above I can quickly get a contradiction and prove FLT. It is neat that it all reduces to one relatively simple equation.

(Maybe I should mention that 'q' comes from (x+y-z)^p = p(z-x)(z-y)x+y)Q, where Q = q^p, which comes from the identity, (x+y-z)^p = x^p + y^p - z^p + p(z-x)(z-y)(x+y)Q, which the mathematicians assure me has been known since well before Euler. That's ok, since I found it and proved it myself before they told me. Nothing here is in dispute.)

Why do I think a physics background was extremely helpful (and maybe necessary) in solving this old "pure" math problem?

Because physics is about challenging what is known, while mathematics is about building on what is known with the assumption that the foundation is complete. From my experience, this viewpoint remains despite Godel's proof.

I don't care how many people look for something. Just because they can't find it, you can't just believe it's not there. This statement doesn't bother a physicist but it goes against the grain of a mathematician.

I think it's time they had a paradigm shift.

In case you're curious after all of that the website is

http://home.earthlink.net/~jharris2/FLTb.htm

I got an email today from a mathematician with a question about an important statement in my proof of FLT. I replied with the answer and realized that I needed to show a few more steps in my proof since I thought the statement was pretty obvious, but the guys a Ph.d so I guess it's not. In any event, I've updated the website and the proof. It's still correct.

Also, I've debated removing the section where I prove Case 1 or thinking maybe I should put it as an addendum or something since it's not necessary for the complete proof and I think people are getting stuck there.

I've decided to leave everything as is, but I'll say this, you can skip that section because what follows proves Case 1 and Case 2 in one swoop.

In case anyone cares, here's the rest of the story. If you decide to wade through what follows (it's not really long), I'll tell you why my proof is correct, why most of you can evaluate it without much difficulty, and possibly why it took someone with a physics background to solve it simply.

I've been looking for a simple proof of FLT since the one given by Wiles' because intuition told me if the thing was true then there was a relatively simple reason. I figured I wouldn't need much more math than I already knew and that I didn't want to go over old (and failed trails) that had been produced in the over 350 years mathematicians have been looking at the problem. But, I knew I would need some help.

It's not the easiest thing to get help or attention from mathematicians on the subject. Seems "amateurs" have been coming at them with proofs for just about all of the 350 years the problem has been around. In fact, some of them can get downright testy about it. So I've been creative. Over the past three years I've made a very public search on sci.math with a decent amount of help. I've also often claimed success only to have to recant later when error(s) were found.

So, the readers of sci.math (and possibly now the readers of sci.physics) have good reason to doubt me other than because of the very strong belief that something simple just *couldn't* have been missed for all of this time.

Now, I get to tell you why my proof is true. On my first webpage I introduce the problem and show why I work on it as x^p + y^p = z^p, where p is an odd prime, and I assume integer solutions x,y,z so that I can prove the theorem by contradiction. On my proof page (the third one), I rewrite this in terms of u,y,k and u,y,f instead of x,y,z (pretty basic at this point), where u = fk, and u and y are integers if x,y and z are.

The neat thing is that the polynomials created by this rewrite have roots for f and k that are limited in a special way. I can easily prove that each set has exactly two real roots with the rest imaginary (complex).

Since I can relate each root from the polynomial using u = fk, I know that the sum of the complex roots for one of the polynomials is just a multiple of the sum of the complex roots of the other.

That's the astounding thing that makes everything else possible, because using that fact, I can relate the first coefficient of one polynomial with the first of the other, since it just equals the sum of the roots. So, I relate the moduli of the sums relative to a number I haven't mentioned yet, which I call "q".

And the result of all this effort is just the relation, z^2 = xy(mod q).

That's as far as I go here because I don't want a really long post. Suffice it to say that using the above I can quickly get a contradiction and prove FLT. It is neat that it all reduces to one relatively simple equation.

(Maybe I should mention that 'q' comes from (x+y-z)^p = p(z-x)(z-y)x+y)Q, where Q = q^p, which comes from the identity, (x+y-z)^p = x^p + y^p - z^p + p(z-x)(z-y)(x+y)Q, which the mathematicians assure me has been known since well before Euler. That's ok, since I found it and proved it myself before they told me. Nothing here is in dispute.)

Why do I think a physics background was extremely helpful (and maybe necessary) in solving this old "pure" math problem?

Because physics is about challenging what is known, while mathematics is about building on what is known with the assumption that the foundation is complete. From my experience, this viewpoint remains despite Godel's proof.

I don't care how many people look for something. Just because they can't find it, you can't just believe it's not there. This statement doesn't bother a physicist but it goes against the grain of a mathematician.

I think it's time they had a paradigm shift.

In case you're curious after all of that the website is

http://home.earthlink.net/~jharris2/FLTb.htm

I got an email today from a mathematician with a question about an important statement in my proof of FLT. I replied with the answer and realized that I needed to show a few more steps in my proof since I thought the statement was pretty obvious, but the guys a Ph.d so I guess it's not. In any event, I've updated the website and the proof. It's still correct.

Also, I've debated removing the section where I prove Case 1 or thinking maybe I should put it as an addendum or something since it's not necessary for the complete proof and I think people are getting stuck there.

I've decided to leave everything as is, but I'll say this, you can skip that section because what follows proves Case 1 and Case 2 in one swoop.

### Wednesday, May 12, 1999

## Mathematicians can't find error, but deny reality

Having a physics degree I thought I'd post a link to my proof of FLT here as well as on sci.math because it doesn't take a mathematician to evaluate it. It turns out that the problem only requires a little bit of complex analysis and some modular aritmetic (like 17 = 2(mod 5) i.e. 17 - 5*3 = 2)

The link to this proof has been presented on sci.math for almost a month. The website has had 399 visitors to date. Primarily from sci.math. However, the sci.math' ers haven't been able to find a significant error (they found a grammar mistake which I corrected and a sign error which I corrected).

Yet, I get email that I couldn't have a proof because it's not possible as 360 years of mathematicians couldn't have missed that. On the newsgroup the person who accepted a challenge from me to find an error to the proof now writes limericks about me (he failed the challenge). In fact, if you go to sci.math you'll see several posts (including lots of limericks) about "JSH" all disdainful, and none with a comment on the math of the proof.

Now I have to present it to a math journal and wait six months to a year for a reply. Always with the likelihood that the journal would refuse to consider what I present, and I'm supposed to present to one journal at a time. It took four years to find the proof. It's looking like it could take as long to get it reviewed.

So, as I get started on this process, I thought it wouldn't hurt to put the link here. The website will be at this link for about a week before I pull the site as I'm changing providers and email address soon.

http://home.earthlink.net/~jharris2/FLTb.htm

The link to this proof has been presented on sci.math for almost a month. The website has had 399 visitors to date. Primarily from sci.math. However, the sci.math' ers haven't been able to find a significant error (they found a grammar mistake which I corrected and a sign error which I corrected).

Yet, I get email that I couldn't have a proof because it's not possible as 360 years of mathematicians couldn't have missed that. On the newsgroup the person who accepted a challenge from me to find an error to the proof now writes limericks about me (he failed the challenge). In fact, if you go to sci.math you'll see several posts (including lots of limericks) about "JSH" all disdainful, and none with a comment on the math of the proof.

Now I have to present it to a math journal and wait six months to a year for a reply. Always with the likelihood that the journal would refuse to consider what I present, and I'm supposed to present to one journal at a time. It took four years to find the proof. It's looking like it could take as long to get it reviewed.

So, as I get started on this process, I thought it wouldn't hurt to put the link here. The website will be at this link for about a week before I pull the site as I'm changing providers and email address soon.

http://home.earthlink.net/~jharris2/FLTb.htm

### Tuesday, May 04, 1999

## JSH: Webpage statistics

As of today my webpage with a simple proof of FLT has been visited 365 times.

Without the detailed stats that I got with my commercial site, it's impossible to tell how many of those were unique.

At least 30 of those visits were from me checking the count. That number also doesn't reflect those who went directly to the proof on page 3 which doesn't update the counter.

As a conservative estimate (trying to exclude robot programs and re-visits) I'd guess at about 100 unique visitors.

Of that 100 maybe 20 actually could follow the math if they chose to.

From posts or emails, I know of only three people who have looked at the site carefully.

From all of this, the only errors found were minor and were fixed as soon as they were pointed out (I'd put 'y' instead of 'y^p' in one place and I had a sign wrong on the first "Background" page).

Besides that I found two serious errors which invalidated the proof, though no one else noticed, and I fixed them. That was four days or so ago. So, there is now a proof at the site.

At least one person has taken it upon himself to find errors but I haven't heard from him or noticed any posts by him recently.

To make sure that someone else doesn't take credit, I've emailed the correct webpage to several people who will keep it for me. That was done several days ago. Due to the very public nature of my search, I'm not that worried about credit, but you can't be too careful.

Based on all the evidence I have before me now, I feel confident in concluding that I have a proof. Sci.math has played its role well as always.

Without the detailed stats that I got with my commercial site, it's impossible to tell how many of those were unique.

At least 30 of those visits were from me checking the count. That number also doesn't reflect those who went directly to the proof on page 3 which doesn't update the counter.

As a conservative estimate (trying to exclude robot programs and re-visits) I'd guess at about 100 unique visitors.

Of that 100 maybe 20 actually could follow the math if they chose to.

From posts or emails, I know of only three people who have looked at the site carefully.

From all of this, the only errors found were minor and were fixed as soon as they were pointed out (I'd put 'y' instead of 'y^p' in one place and I had a sign wrong on the first "Background" page).

Besides that I found two serious errors which invalidated the proof, though no one else noticed, and I fixed them. That was four days or so ago. So, there is now a proof at the site.

At least one person has taken it upon himself to find errors but I haven't heard from him or noticed any posts by him recently.

To make sure that someone else doesn't take credit, I've emailed the correct webpage to several people who will keep it for me. That was done several days ago. Due to the very public nature of my search, I'm not that worried about credit, but you can't be too careful.

Based on all the evidence I have before me now, I feel confident in concluding that I have a proof. Sci.math has played its role well as always.