### Saturday, March 28, 1998

## Fallacious Logic and latest news

Logic in no way dictates that because I have many false proofs that I can't have a correct one now. That's why I present it to see if it can be shot down (and so far it hasn't, I just keep getting increasingly rude comments).

Neither does logic dictate that because a simple proof hasn't been public before now that one doesn't exist. A point admitted by Professor Wiles.

And, of course, simply because most mathematicians believe there isn't a simple proof. logic doesn't say that is true.

Mathematics couldn't care less what people think.

Obviously, logic does dictate that no simple solution could be found if no one looked for it.

The proof is not unlooked at. Part of my strategy has always been to be in contact with those who have the ability an inclination to consider such things. I am not only dependent on the comments posted on newsgroups.

Neither does logic dictate that because a simple proof hasn't been public before now that one doesn't exist. A point admitted by Professor Wiles.

And, of course, simply because most mathematicians believe there isn't a simple proof. logic doesn't say that is true.

Mathematics couldn't care less what people think.

Obviously, logic does dictate that no simple solution could be found if no one looked for it.

The proof is not unlooked at. Part of my strategy has always been to be in contact with those who have the ability an inclination to consider such things. I am not only dependent on the comments posted on newsgroups.

### Sunday, March 22, 1998

## The lie of modern mathematics

Dedicated to Brian Hutchins. Maybe he should study a little psychology.

Meaning is given to activities by groups of people. Many mathematicians today seem to believe that they have meaning in their work based on the approval or disapproval of other mathematicians. But as research in "pure mathematics" continues, such opinion becomes increasingly meaningless as fewer and fewer understand what anyone is doing.

Not even mentioning the loss of support from society the opinion of which is disdained anyway by most in the ivory towers of academia. Despite the realities of who pays the bills… It's not impossible to imagine the doll someday being cut off.

But of course, there is an importance in the activity for its own sake, right? That puts mathematical work on the level with the delusions of the insane. It is important to them after all.

Back to the point of the appoval or disapproval of mathematicians themselves. What does it matter when so few have any real understanding of what any particular mathematican is talking about. Complexity is piled upon complexity. Proofs are of incredible length and everyone acts like there's a certainty there. In my opinion, that sounds like delusion.

It all comes back to a certain arrogance. As if God ordained a task so grand. But Mathematics is infinite and has real value to us based on its usefulness to and in our finite little world. At least Godel sucked some of the air out of such hubris. So many are trekking off into infinite with little understanding. Trampling in their haste over so much that is certainly of value in more limited circles. Has everything simpler really been that well explored?

Ultimately, I think that much of what we call "modern mathematics" will simply die off. Endless journals and books filled with what may as well be mindless doodling because eventually no one will ever read any of it. Until then, I do thank God that you mainly burden yourselves with such because most of it is already being ignored by the mainstream…unless it is of use to them.

Which is as it should be. "Pure mathematics" research sounds to me like just another form of social welfare. And those that hide behind the term, sound too afraid to accept the challenge of producing something of real value to either those in the present or near future.

Just some thoughts that have been percolating. Having been diagnosed as gifted at a young age, I've grown up fighting the tendencies in myself and about of deifying intellect. I started learning calculus at twelve. Learned a years worth of geometry in six days at Duke University (summer program for gifted youth). Ttaught myself trigonometry in three and was studying partial differentials when I was fifteen. My entire life I've been in the top 2% in terms of overall intelligence and in the 99th percentile with mathematics based on standardized tests.

So what. And look what good it's done me. There is a lot more to life.

Meaning is given to activities by groups of people. Many mathematicians today seem to believe that they have meaning in their work based on the approval or disapproval of other mathematicians. But as research in "pure mathematics" continues, such opinion becomes increasingly meaningless as fewer and fewer understand what anyone is doing.

Not even mentioning the loss of support from society the opinion of which is disdained anyway by most in the ivory towers of academia. Despite the realities of who pays the bills… It's not impossible to imagine the doll someday being cut off.

But of course, there is an importance in the activity for its own sake, right? That puts mathematical work on the level with the delusions of the insane. It is important to them after all.

Back to the point of the appoval or disapproval of mathematicians themselves. What does it matter when so few have any real understanding of what any particular mathematican is talking about. Complexity is piled upon complexity. Proofs are of incredible length and everyone acts like there's a certainty there. In my opinion, that sounds like delusion.

It all comes back to a certain arrogance. As if God ordained a task so grand. But Mathematics is infinite and has real value to us based on its usefulness to and in our finite little world. At least Godel sucked some of the air out of such hubris. So many are trekking off into infinite with little understanding. Trampling in their haste over so much that is certainly of value in more limited circles. Has everything simpler really been that well explored?

Ultimately, I think that much of what we call "modern mathematics" will simply die off. Endless journals and books filled with what may as well be mindless doodling because eventually no one will ever read any of it. Until then, I do thank God that you mainly burden yourselves with such because most of it is already being ignored by the mainstream…unless it is of use to them.

Which is as it should be. "Pure mathematics" research sounds to me like just another form of social welfare. And those that hide behind the term, sound too afraid to accept the challenge of producing something of real value to either those in the present or near future.

Just some thoughts that have been percolating. Having been diagnosed as gifted at a young age, I've grown up fighting the tendencies in myself and about of deifying intellect. I started learning calculus at twelve. Learned a years worth of geometry in six days at Duke University (summer program for gifted youth). Ttaught myself trigonometry in three and was studying partial differentials when I was fifteen. My entire life I've been in the top 2% in terms of overall intelligence and in the 99th percentile with mathematics based on standardized tests.

So what. And look what good it's done me. There is a lot more to life.

### Friday, March 20, 1998

## After 3yrs a search ends

As certain as I am of the success in this latest method, I'm more certain that I'm finished with the problem. I know that I've learned quite a bit about myself and a lot about others; although, I don't yet know what was the point.

Most of the time lately, I've simply said that I'm a child of Destiny. In many ways, it feels like I was born to such a task. But, of course, why wouldn't it.

Maybe it's more that much of my life is just staying busy while either waiting for the point or the end. This task has kept me somewhat busy.

I have often thought of all those who've worked on this problem—many of them in secret. Pouring in their time, their effort…and thus their lives. In the end, I think that's what can really gives it all meaning. The nobleness of any task being given to it by the people who sweat blood and tears; giving their passion and heart.

Here's to all of them. Here's to me. And here's to you. May we all find our nobility.

Most of the time lately, I've simply said that I'm a child of Destiny. In many ways, it feels like I was born to such a task. But, of course, why wouldn't it.

Maybe it's more that much of my life is just staying busy while either waiting for the point or the end. This task has kept me somewhat busy.

I have often thought of all those who've worked on this problem—many of them in secret. Pouring in their time, their effort…and thus their lives. In the end, I think that's what can really gives it all meaning. The nobleness of any task being given to it by the people who sweat blood and tears; giving their passion and heart.

Here's to all of them. Here's to me. And here's to you. May we all find our nobility.

### Monday, March 16, 1998

## Proof of FLT? You be the judge.

Why try? Why not. I've been looking over what should have been old ground for a simple proof because Fermat said it was there; my intuition says it's there; and because I've felt like it (it has been fun most of the time).

Problem:

Given x^p + y^p = z^p; x,y,z relatively prime; p odd prime; show no solution exists.

My current method (which hasn't been disproven as of yet) is to take

(s+a)^p + (s+b)^p = (s+c)^p

as a subs. for x^p + y^p = z^p.

Then I choose a,b,c such that a+b=c which interestingly means that

s = x+y-z. [(a=z-y); (b=z-x); (c=2z-(x+y))]

I then expand and group and also notice that a^p + b^p - c^p = -pabcQ (Q is just an integer to handle the general expansion, for p=3, Q=1).

I then note that the eqn thus found must be of the form (s+A)(s+B)(s+C)...=0

Finally, I notice that c=2z-(x+y)=z + z-(x+y) = z - s, and that this reduces the eqn thus formed from a highest s of s^p to s^{p-1} (you have to be working with the expansion to see this)

But notice that the one of the roots must be divisible by z and I have a contradiction because the roots of the original aren't.

The above shouldn't make a lot of sense because I'm in a bit of a hurry and am pushing out the conclusion. I would recommend that you do two things to get an idea of what I'm doing.

1. Try it with x^2 + y^2 = z^2 (which gives s^2 + a^2 + b^2 - c^2 = 0, note that

(a+b-c)^2 = a^2+b^2-c^2 + 2ab)

2. Then try it with p=3 because the expansions are simpler like

(a+b-c)^3 = a^3+b^3 - c^3 + 3abc

The method is a bit strange even to me but it seems to work. It leaves the question to later (once it has been verified to work by others) as to why it wasn't found earlier.

Get through the above and you will be one of the few people who currently know the simple explanation of why the theorem is true. After all, Wile's proof is a bit complex.

Problem:

Given x^p + y^p = z^p; x,y,z relatively prime; p odd prime; show no solution exists.

My current method (which hasn't been disproven as of yet) is to take

(s+a)^p + (s+b)^p = (s+c)^p

as a subs. for x^p + y^p = z^p.

Then I choose a,b,c such that a+b=c which interestingly means that

s = x+y-z. [(a=z-y); (b=z-x); (c=2z-(x+y))]

I then expand and group and also notice that a^p + b^p - c^p = -pabcQ (Q is just an integer to handle the general expansion, for p=3, Q=1).

I then note that the eqn thus found must be of the form (s+A)(s+B)(s+C)...=0

Finally, I notice that c=2z-(x+y)=z + z-(x+y) = z - s, and that this reduces the eqn thus formed from a highest s of s^p to s^{p-1} (you have to be working with the expansion to see this)

But notice that the one of the roots must be divisible by z and I have a contradiction because the roots of the original aren't.

The above shouldn't make a lot of sense because I'm in a bit of a hurry and am pushing out the conclusion. I would recommend that you do two things to get an idea of what I'm doing.

1. Try it with x^2 + y^2 = z^2 (which gives s^2 + a^2 + b^2 - c^2 = 0, note that

(a+b-c)^2 = a^2+b^2-c^2 + 2ab)

2. Then try it with p=3 because the expansions are simpler like

(a+b-c)^3 = a^3+b^3 - c^3 + 3abc

The method is a bit strange even to me but it seems to work. It leaves the question to later (once it has been verified to work by others) as to why it wasn't found earlier.

Get through the above and you will be one of the few people who currently know the simple explanation of why the theorem is true. After all, Wile's proof is a bit complex.

### Monday, March 02, 1998

## Before I write FLT proof one more post

Because it'll take me a bit of time to write a satisfactory proof (assuming I can do so), I thought I'd make one more post outlining the method and the evidence that makes me so certain now that I have proven Fermat's Last Theorem.

As I've stated before, x^p + y^p = z^p can be rewritten as

(x+y-z)^p = p(z-x)(z-y)(x+y)Q

I have to introduce the Q term because of the complexity of the general expansion. Also, I continue to use a positive z because I get signs confused when I use

x^p + y^p + z^p = 0

Now, it's obvious enough in the above that the right and left must be divisible by 2 raised to the same power. Let's assume that x is even. Then (z-y) must be divisible by 2 raised to a power that is divisible by p.

Now then, what if I write

(x+ay^b - z)^p = y^p[(ay^{b-1})^p - 1] + p(z-x)(z-ay^b)(x+ay^b)Q{a,b}

It should be obvious to anyone that I can find an 'a' and 'b' such that (z-ay^b) is divisible by 2 raised to the same power as (z-y) i.e. if (z-y)=2^s F then I can have (z-ay^b)=2^s G

Well that's all well and good unless I tell you that I can make y^p[(ay^{b-1})^p - 1] also divisible by 2^s. (The sum of the two terms is then divisible by 2^{s+1})

To do so, all that is necessary is that I show that x must be divisible by a 2 raised to a higher power than y-1 or z-1 are. i.e. if y-1 is divisible by 4 and not 8 then x must be divisible by at least 8.

And that is all that is necessary to prove Fermat's Last Theorem.

Of course, n=2 must not be invalidated by the above.

Consider, x^2 + y^2 = z^2 which is the same as (x+y-z)^2 = 2(z-x)(z-y)

Using the same method as above, I'd have

(x+ay^2 - z)^2 = y^2[(ay)^2 - 1] + 2(z-x)(z-ay^2)

In no case is (y-1) allowed to be divisible by a higher power of 2 than x unless (z-y) is divisible by 2 and not by 4 (a possibility not open to p odd prime).

For instance, 13^2 + 84^2 = 85^2. Notice that 13-1 is divisible by 4 at most as is 84.

Why is this? To prevent what proves FLT. If the even term is divisible by a greater power of two than the odd minus 1 then there comes a gap through which a contradiction must follow.

Astute readers can quickly figure out why. I would just as soon explain it in detail but it would make too long of a post. That's where all the legwork I've done over the past couple of years comes in. Basically, all I do is show that z+x=(z-y)+(x+y) trivial as it may seem helps prove that the even term must be divisible by a greater power of 2 than either odd minus 1. And that by writing x^p = (z-y)(z^{p-1}+…+y^{p-1}).

That much is simple and hopefully many of you understand it from that. In a complete proof it's necessary to fill in the details.

Last thing. Above I have an 'a' and 'b' which I say can be chosen such that (z-ay^b) is still divisible by the same powers of 2 as (z-y) as is [(ay^{b-1})p -1]. To do so, I only need b = 2^s where s>t where t is (z-y)=2^t F.

That actually makes this post a proof but not in the form most mathematicians would consider complete. Oh well. I guess that means I have work to do.

As I've stated before, x^p + y^p = z^p can be rewritten as

(x+y-z)^p = p(z-x)(z-y)(x+y)Q

I have to introduce the Q term because of the complexity of the general expansion. Also, I continue to use a positive z because I get signs confused when I use

x^p + y^p + z^p = 0

Now, it's obvious enough in the above that the right and left must be divisible by 2 raised to the same power. Let's assume that x is even. Then (z-y) must be divisible by 2 raised to a power that is divisible by p.

Now then, what if I write

(x+ay^b - z)^p = y^p[(ay^{b-1})^p - 1] + p(z-x)(z-ay^b)(x+ay^b)Q{a,b}

It should be obvious to anyone that I can find an 'a' and 'b' such that (z-ay^b) is divisible by 2 raised to the same power as (z-y) i.e. if (z-y)=2^s F then I can have (z-ay^b)=2^s G

Well that's all well and good unless I tell you that I can make y^p[(ay^{b-1})^p - 1] also divisible by 2^s. (The sum of the two terms is then divisible by 2^{s+1})

To do so, all that is necessary is that I show that x must be divisible by a 2 raised to a higher power than y-1 or z-1 are. i.e. if y-1 is divisible by 4 and not 8 then x must be divisible by at least 8.

And that is all that is necessary to prove Fermat's Last Theorem.

Of course, n=2 must not be invalidated by the above.

Consider, x^2 + y^2 = z^2 which is the same as (x+y-z)^2 = 2(z-x)(z-y)

Using the same method as above, I'd have

(x+ay^2 - z)^2 = y^2[(ay)^2 - 1] + 2(z-x)(z-ay^2)

In no case is (y-1) allowed to be divisible by a higher power of 2 than x unless (z-y) is divisible by 2 and not by 4 (a possibility not open to p odd prime).

For instance, 13^2 + 84^2 = 85^2. Notice that 13-1 is divisible by 4 at most as is 84.

Why is this? To prevent what proves FLT. If the even term is divisible by a greater power of two than the odd minus 1 then there comes a gap through which a contradiction must follow.

Astute readers can quickly figure out why. I would just as soon explain it in detail but it would make too long of a post. That's where all the legwork I've done over the past couple of years comes in. Basically, all I do is show that z+x=(z-y)+(x+y) trivial as it may seem helps prove that the even term must be divisible by a greater power of 2 than either odd minus 1. And that by writing x^p = (z-y)(z^{p-1}+…+y^{p-1}).

That much is simple and hopefully many of you understand it from that. In a complete proof it's necessary to fill in the details.

Last thing. Above I have an 'a' and 'b' which I say can be chosen such that (z-ay^b) is still divisible by the same powers of 2 as (z-y) as is [(ay^{b-1})p -1]. To do so, I only need b = 2^s where s>t where t is (z-y)=2^t F.

That actually makes this post a proof but not in the form most mathematicians would consider complete. Oh well. I guess that means I have work to do.

## The Method behind my Madness

Obviously, it doesn't serve someone who wants to be taken seriously to make outlandish posts every couple of months. But what if there were a simple explanation?

Imagine you were engaged in what many believed to be a pointless search. All of the experts said that you couldn't succeed and they were all annoyed by endless amateur efforts in the area. Add to that an accepted proof that seemed to finish it all.

After all, what's the point for another proof when one has been found?

Unless, every one of your instincts tell you that there is a far simpler solution than most believe.

Why is that important?

Because it says things about mathematics, about human thought, about meta-mathematics, about metaphysics, and it makes a statement that can echo through history.

When are we really certain? When is something known? When all of the experts say it is?

With all of the excess baggage, the search has been a strange one. I've made outlandish posts claiming success whenever I had a significant result. Why? Maybe I wanted to see if I could actually get someone's attention…if anyone of capability would ever critique a finished work. Maybe mostly it was just incautious exuberance. A lot of times I was just tired and wanted help. And, maybe I wanted to just test how I would react and whether or not I could handle the consequences.

Now, I finally have a solution and I have to figure out how best to do this. I will write up a complete proof. There's no reason not too. And I don't need any help in doing so. For anyone who cares, it is mostly outlined in my previous 'evidence' post. Little details can be filled in from earlier posts, though, it should be obvious from it.

In the end, the community of the Internet newsgroups will have a deciding factor in the timing of the revelation of my discoveries. I guess I care a bit or I wouldn't be producing this post. If necessary, I have other avenues; although, they will take time.

Maybe it isn't important to most of you for there to be a simple solution. But I doubt it. I wonder how many of you will go with your own instincts and keep an open mind about the possibility.

Imagine you were engaged in what many believed to be a pointless search. All of the experts said that you couldn't succeed and they were all annoyed by endless amateur efforts in the area. Add to that an accepted proof that seemed to finish it all.

After all, what's the point for another proof when one has been found?

Unless, every one of your instincts tell you that there is a far simpler solution than most believe.

Why is that important?

Because it says things about mathematics, about human thought, about meta-mathematics, about metaphysics, and it makes a statement that can echo through history.

When are we really certain? When is something known? When all of the experts say it is?

With all of the excess baggage, the search has been a strange one. I've made outlandish posts claiming success whenever I had a significant result. Why? Maybe I wanted to see if I could actually get someone's attention…if anyone of capability would ever critique a finished work. Maybe mostly it was just incautious exuberance. A lot of times I was just tired and wanted help. And, maybe I wanted to just test how I would react and whether or not I could handle the consequences.

Now, I finally have a solution and I have to figure out how best to do this. I will write up a complete proof. There's no reason not too. And I don't need any help in doing so. For anyone who cares, it is mostly outlined in my previous 'evidence' post. Little details can be filled in from earlier posts, though, it should be obvious from it.

In the end, the community of the Internet newsgroups will have a deciding factor in the timing of the revelation of my discoveries. I guess I care a bit or I wouldn't be producing this post. If necessary, I have other avenues; although, they will take time.

Maybe it isn't important to most of you for there to be a simple solution. But I doubt it. I wonder how many of you will go with your own instincts and keep an open mind about the possibility.