Sunday, August 30, 2015

Actual value of recognition

To some extent the way attention works for something you find is easy--other people learn of it and use it.

The more people who learn of it and use it, the more influence some thing you found can have.

The true value of recognition by someone of your work I think is in referral, which can happen a couple of ways as to how someone refers things to someone else. Usually it's by mentioning it to someone, or using something and having someone ask about it. That latter is a BIG driver. Human curiosity can demand to know--how did you do that?

The web facilitates referral.

Am confident that most of my influence from my mathematical discovery happens because someone needs the mathematics.

Which is why I often focus on some math of mine that fulfills a real world need, as then I can explain why it gets the functional recognition that drives it around the world. And the web gives me objective measures to show that happened.

The web simplifies distribution of information. It is an information distribution mechanism.

James Harris

Sunday, August 23, 2015

My trust in math

Maybe I should get serious and find math software that I can recommend to people for when I talk about how computers are my friends as they can easily verify some my math.

Mathematical discovery has the awesome property that a correct result is immutable. That's a fancy word for unchanging. So you can chill out, relax, sit back and enjoy the ride if you have one. But for other people looking on it can be stressful--how can they know? Sure you can say, just work through the proof, but realistically how many people really want to do that work before they have confidence you are correct?

So I've been working on what I call the social problem, which includes ways to help, like pointing out that computers verify certain really cool things I found perfectly and easily.

Checking by computer is quick and easy if you know your way around math software.

That social stuff can mess with people's minds and I believe quickly checkable information helps, so I've talked more in such areas recently, like my time on the Usenet newsgroup sci.math and other things that for the most part are really about perception.

How might it look to someone just coming across these things who doesn't have over a decade of experience playing with the math?

To me numbers are just FUN. So I at times I find myself stuck thinking: why can't I just show them the fun? Don't they get it?

Like, one of my favorite math expressions now:

(462 + 482 + 722)(1722 + 258+ 430+ 6022 +17622) = 

            615+ 30752 + 141452 + 159902  + 1884972   =  114*74*210

That relies on some math I found to get it. As I can do lots with squares and sums of squares to get squares. Like I posted recently to emphasize a way to get multiple sums of squares for the same value, which might be useful to math hobbyists.

What other people do with the discoveries, especially if they find new math of their own is what will shift the social reality.

What changes things is what others find as that can make it really real to a person.

Like, long ago, I just dreamed of having my own mathematical discoveries and wondering what it would feel like.

It's like--the math is your best friend EVER.

The math will never let you down. The math will never betray you. The math will never change on you.

If you find out something was wrong, it was people who screwed up, not the math. That's a weird thing to contemplate which I love so much. Guarantee of human error when it doesn't work!

Put your trust in math, if you can, and then you can have a certainty for which others can only dream.

But you can't manufacture that want. For some people like me it is an aching need.

I had to have that certainty, and knew that mathematics could provide, so I was going to go get it.

The truly motivated will figure it out and take the time required. But to do the work you have to believe in the math. As otherwise why would you do what it takes?

James Harris

Wednesday, August 19, 2015

How computers help me

Our world shifted a lot very rapidly and my story is very much a web story, where computers around the globe endlessly help me, and without them, well you wouldn't be reading these words.

And it's not just about computers and the web allowing me to communicate my mathematical findings, but about them making checking trivial.

So for instance I've decided to focus lately on a better way to reduce binary quadratic Diophantine equations, which doesn't even care about the discriminant.

Well, turns out you can just plug in my way into math software--which I've never done--and it will just check it for you, and for instance solve for a variable I call 's'.

Here's me doing the calculation with a simple example. You can see how it could be difficult with the general case.

If someone plugs my method for reducing into the appropriate math software it gives s, as a function of x and y, which I know because years ago, back in 2008 I think, some people did it.

I've posted about my years on Usenet, where most was on the sci.math newsgroup which some people seem to think is significant from search results I've seen. For me it was a way to get things checked, and I've concluded that math people there thought I looked stupid because of some of the things I was trying. And for them, showing errors in what I was doing was proving my stupidity! So they'd fall all over themselves critiquing my ideas, which often I'd simply brainstorm. So I could generate the ideas meaning credit was always mine alone, while they'd simply critique! Often insultingly, which I don't condone, but it's not like I could control it either. That was their choice. To me it would be easy enough to point out an error without also having to insult a person.

Because it is an example where computers make checking easy, it means there's little effort in some math student verifying. Weirdly enough, if you know the history in this area I can take it up a notch and note that I did in fact improve on Gauss. The computer does ALL the real work. All that student needs to do is know how to use it, and just type in the equations and watch it whirl.

And Gauss is a hero of mine, so that's a big deal to me. And I do not say it lightly.

Actually I'm sure I overrated his importance to the modern math field as for a while I thought that would be it! No way something so huge could be ignored. But mathematicians today are more focused on Riemann I think who came after Gauss.

With such a claim it is a relief for me that any person with a computer and math software who knows how to use it can verify. Computers back me up.

While I wouldn't mind established mathematicians validating my research, this way is more fun. And is less work for me! Besides we're in a fascinating new situation with the arrival of the web, and wouldn't it have been so much more boring without a nice dramatic story like mine?

Why does history seem to require such?

Reality is I'd prefer computers to validate me than mathematicians. Wouldn't you?

Human beings can just be wrong.

And computers have made this story possible.

Puzzling these things out helps me I think and maybe others.

Computers let me reach the world, validate my research, and are tireless in that defense.

I'd rather be validated by computers.

James Harris

Sunday, August 16, 2015

Approximating square root of three

Sometimes I figure things out just for the fun of it.

When I get bored I can play with some math of mine. And noticed this thing years ago, but don't think I've put it out there.

With my BQD Iterator let F = 1, u=x, v=y, and D = -3. Then:

1. x2 - 3y2 = 1

2. (x+3y)2 - 3(x+y)2 = -2

3. (4x+6y)2 - 3(2x + 4y)2 = 4

Divide off 4, and you have: (2x+3y)2 - 3(x + 2y)2 = 1

4. (5x + 9y)2 - 3(3x + 5y)2 = -2

5. (14x + 24y)2 - 3(8x + 14y)2 = 4

Divide off 4 again, and you get: (7x + 12y)2 - 3(4x + 7y)2 = 1

And x = 1, y = 0 works, so I have:

1. 12 - 3(0)2 = 1

2. (1)2 - 3(1)2 = -2

3. (2)2 - 3(1)2 = 1

4. (5)2 - 3(3)2 = -2

5. (7)2 - 3(4)2 = 1

and you can keep going out to infinity.

And you're getting approximations to square root of 3, with the ratio of x/y, like 7/4 = 1.75.

But wow that is SLOW.

Oh yeah, you can use the algebraic ones to move things along as well. So let's plug the last result of x = 7, y =4, into the last algebraic one: (7x + 12y)2 - 3(4x + 7y)2 = 1

(7*7 + 12*4)2 - 3(4*7 + 7*4)2 = 1

(97)2 - 3(56)2 = 1

And is approximately 1.732 to three significant figures. And yes, you can now plug x = 97 and y = 56 back into it, if you wish to continue. Or you could move further out with an algebraic solution. So yeah there are an infinity of such equations to generate an approximate square root of 3. Guess that's kind of cool.

I'm curious! So I'll plug them back into what I have. Not quite curious enough to extend to the next algebraic one:

(7*97 + 12*56)2 - 3(4*97 + 7*56)2 = 1

(1351)2 - 3(780)2 = 1

So I now have 1351/780 which is approximately 1.73205, which I think is more satisfying.

You can also look at 13512/7802 is approximately 3.000001.

That was fun! And it was something quick to do.

I LOVE to play with actual numbers. My discoveries mean I can always get my number theory fix whenever I need it. Often I'll figure something out just to feel better.

Works marvelously for some reason.

James Harris

Thursday, August 13, 2015

Considering modular with a cubic Diophantine

Spend a lot of time with quadratics but also did extension of some ideas to the cubic equation:

x3 - Dy3 = F

And found that I could solve modulo N, where N is a cubic residue of D that does not share any prime factors with F, m is the cubed residue, so: m3 = D mod N, and r is some residue modulo N, where Fr-1 exists.


3(2my+r)2  = 4Fr-1 - r2  mod N


x = my + r mod N

Here's a link to derivation of the result. The gist is just factoring x3 - m3y3 = F mod N, in order to get those modular solutions.

Usage steps:

First step then is to find some N for which D is a cubic residue, which does not share prime factors with F, then find m, which is the cubed residue, then for some residue r, you can find y if it exists, and then x.

Some numeric examples:

To explore and confirm equations are correct will not specify F, but will see what I can do knowing just D, so for example if you have D = 11, a close cube is 27, so N = 16 will work if F is odd, and I have then m = 3.

So I have:

3(6y+r)2  = 4Fr-1 - r2  mod 16,

And 3(11) = 1 mod 16, so: (6y+r)2  = 12Fr-1 - 11r2  mod 16

And the only quadratic residues of 16, are: 1, 4 and 9

Can pick any odd r and have its modular inverse exist, so let r =3 to start. Then:

(6y+3)2  = 4F - 3  mod 16, which means:

F = 1, 5 mod 16 for 1, 4 is blocked, and 3, or 7 mod 16 for 9.

Which means F = 1, 3, 5, or 7 mod 16, which is all of the odd residues. Well that didn't tell me much then, but at least equations did work.

Noticed that 33 - 11 = 16, but that was excluded with N = 16, to keep F and N from sharing prime factors. But should be able to find that one with a different N.

So will try an odd N. Another close cube is 64, and 64 - 11 = 53, so N = 53, and m = 4.

3(8y+r)2  = 4Fr-1 - r2  mod 53, and 3(18) = 1 mod 53, so:

(8y+r)2  = 19Fr-1 - 18r2  mod 53.

And can figure out r, if x = 3, y =1, as then x = 4y + r mod 53.

So r = -1 mod 53 = 52 mod 53 = r-1 mod 53.

Quadratic residues of 53 are:

1, 4, 9, 12, 13, 16, 17, 25, 29, 36, 40, 48, 49

(8y+r)2  = 19(16)(-1) - 18  mod 53 = -322 mod 53 = -4 mod 53 = 49 mod 53

Which works with 8y - 1 = 7 mod 53, so y = 1 mod 53.

Lessons learned from examples:

Glad I just calculated an r that would work, as tried from the bottom with r = 2 mod 53, which didn't work at all. But I already knew that not every r would work.

And finding N and m is trivially easy. That's good. You can just find cubes near your D, which gives you m, and then N, with N = m3 - D.

Here the solution for y modulo N involving a quadratic residue means it might be possible to check for existence of solutions.

Proving explicit x and y do not exist would definitely involve checking each residue r available for a given N that fits the rules.

If you try each r modulo N, and find that no y modulo N exists, then of course there are no integer solutions for x and y.

Most of my focus with modular solutions has been on quadratics, but it's worth reminding some of the techniques can go beyond quadratics.

James Harris

Wednesday, August 12, 2015

Finding the discussion

Remarkably, over two decades ago I was some guy who decided as a hobby it would be fun to work on old, hard math problems. Needed to talk things out, and found the Usenet newsgroup sci.math was a place that worked. Math ideas I presented would immediately get critiqued by a large number of people, who would also often hurl insults as well, so it was an odd thing. Chasing math ideas is a LOT easier when others help you find errors.

Of course reality can be tough as you see yourself fail a lot. And you can't have an ego about it, as often you look really silly.

My guess is that one of the deep terrors of many mathematicians is looking stupid!!!

If true, then to mathematicians being insulted for looking stupid may be one of the greatest horrors in life, tapping into their deepest fears. While insulting someone for what is seen as stupidity may feel for them to be deeply empowering.

But if my guess is correct that meant that those Usenet people in attacking my ideas may have felt I was the one who looked stupid, so they'd point out errors to show this perceived stupidity, and then on top of that hurl insults with the help! Which explains their motivation. So wacky.

Can you imagine? As you come up with math ideas, people from all over the world falling all over themselves to evaluate them? Day after day for years?

In contrast this blog has an extraordinary peacefulness to it. Though I was briefly intrigued by a burst of discussion recently on one of my posts.

While I pursued several problems during my Usenet posting phase my time focused on Fermat's Last Theorem garnered the greatest attention. For years various attacks on the problem failed, very publicly, and often spectacularly.

But it was so much fun.

That intense discussion hasn't come back, like so far am struggling on Facebook where I last noted that this blog had visits from 441 cities in 65 countries in the last year. That is, I checked in Google Analytics for the previous 365 days. But the thing is, here it's now just math. It's like take it or leave it. No entertainment. And that IS the difference. So I know how it was done, but I like it better to just be some math.

As for the discussions at their height, if you're curious, much of it is still out there. Search!

Did my best back then to be entertaining, as you DO have to work to keep that kind of attention. I think I did ok. That was work! My hobby really turned into a job of sorts given how much time I put into it.

So I think the quiet here is well earned.

James Harris

Tuesday, August 11, 2015

Infinity and binary quadratic Diophantine equations

Possibility is amazing. How do we know what we don't know? But we can have clues from what things people find that are very surprising. Like one of my discoveries years ago was proof that binary quadratic Diophantine equations are in general connected to an infinity of binary quadratic Diophantine equations where my discovery relies on using an identity.

So I found years ago a way to reduce binary quadratic Diophantine equations, which in general look like:

c1x2 + c2xy + c3y2 = c4 + c5x + c6y

You could reduce to a simpler form, where with my method you do NOT care about the discriminant. Turns out the discriminant is unnecessary when reducing these equations, and 'discriminant' is a technical word familiar to people taught classical methods to reduce.

It being unnecessary to me is really cool! And is one reason my way is simply better as I've discussed recently, because I bring it up routinely.

To put things into some perspective, number theory students who have learned about the discriminant for reducing, can imagine a future where students are not even taught it, in that context, as it's unnecessary. Which it is.

I've tried to usually link to one post for my method to generally reduce, where I use a variable 's', which is a function of x and y, where it was ego that I don't give it there.

Turns out that: s = (c2 - 2c1)x - (c2 - 2c3)y - (c6 - c5)

Which was determined by math software as it's just too freaking hard to figure it out by hand, which I started trying to do a couple of times and never finished, so I wouldn't give the value. Others used the math software I should add. I've decided I was just being silly.

The reduction method gives you a reduced form, but if you use it against a reduced form, it gives back a reduced form, as you can reduce no further, which looks like:

u2 + Dv2 = F

when reduced you get

(u-Dv)2 + D(u+v)2 = F(D+1)

Which I've had for years but recently decided maybe I could jazz up interest by naming it, so call it now a Binary Quadratic Diophantine iterator, or BQD Iterator for short.

Oh yeah, so now we've connected almost all binary quadratic Diophantine equations to an infinity of other ones. Turns out there are some exceptions, but they are trivial cases.

Check out one of my fun things that I did with the result years before I named it:

Let F = 1, u=x, v=y, and D = -2.

1. x2 - 2y2 = 1

2. (x+2y)2 - 2(x+y)2 = -1

3. (3x+4y)2 - 2(2x + 3y)2 = 1

4. (7x + 10y)2 - 2(5x + 7y)2 = -1

5. (17x + 24y)2 - 2(12x + 17y)2 = 1

and you can keep going out to infinity, but I'll stop with 4 iterations.

Notice now you can use the simple case of x=1 and y = 0:

1. 12 = 1

2. (1)2 - 2(1)2 = -1

3. (3)2 - 2(2)2 = 1

4. (7)2 - 2(5)2 = -1

5. (17)2 - 2(12)2 = 1

And you have answers to x2 - 2y2 = 1 and x2 - 2y2 = -1.

Which I like to trot out. To me it's just AWESOME.

I really don't know if that series is in an established number theory text, but it does use my BQD Iterator, which appears to be mine alone, so I wonder how it might. Of course someone could have got lucky, just playing around. But then they'd have no clue why that worked!

So yeah, my way of reducing binary quadratic Diophantine equations just throws away the discriminant. And works on any such which is not trivial. That is, you can find ones where my method doesn't work, but those are easily solvable, and equivalent to a unary case anyway. Like the example I usually give:

x2 + 2xy + y2 = c4 + x + y

is: (x+y)2 - (x+y) - c4 = 0

Having the BEST way to reduce these things is guaranteed attention. Turns out they have practical uses, so there is no doubt that people will find this method and use it.

And since the method was discovered with an identity there is no doubt about correctness.

It is perfect. Absolutely perfect.

That an identity can be used in this way is fascinating to me. Finding the identity requires more advanced techniques which to me is just more modular algebra symbology.

If I had done nothing else in my life just this discovery alone would give me my place in mathematical history. So it's like, I'm good. And of course got LOTS of other things, many of which are much, MUCH bigger. So I'm like, chill.

If you believe in mathematics then it's a simple situation.

So I found something cool, but who knows what someone else might find? We don't know what we don't know, you know?

Mathematics is an infinite subject. To me it is a no-brainer that it has an endless ability to surprise us, keeping the spirit of adventure alive. We can never fully encompass the infinity that is Mathematics.

The web links to my pages but years ago I changed the name of this blog, and the web promptly linked to pages on Usenet where I'd discussed it. Until the blog regained position in web search under its new name of Some Math, and the web links switched back!

Our world is hungry for new useful knowledge. It will do what it takes to have and keep it.

That fascinates me so discussed it more than once, where here is a highly analytical post.

But even without them I'm sure plenty of people all over the world have copied the method down, so it's a permanent part of the world's body of knowledge.

And with it comes the stunning reality that in general binary quadratic Diophantine equations are connected to an infinity of other ones, like some massively huge freaking number theoretic family.

You can't stop the connection. It rules.

James Harris

Friday, August 07, 2015

Finding primes using non-square residues

Back in 2007, was probably just trying to come up with something, and put forward an idea for finding primes by using non-square residues. The idea is simple enough.

Given a known prime p, use one of its non-quadratic residues r, with:

p2 - r

and factor the result, to get a new larger prime in it. If a new large one is in it.

At that time I used 29 and 17.

The quadratic residues of 29 are:

1, 4, 5, 6, 7, 9, 13, 16, 20, 22, 23, 24, 25, 28

So yeah, 17 is not one of them.


292 - 17 = 824 = 8*103

Next iteration:

1032 - 17 = 32*331

Second iteration:

3312 - 17 = 8*13693

Third iteration:

136932 - 17 = ( 23 )( 19 )( 43 )( 28687 )

Fourth iteration:

286872 - 17 = ( 24 )( 1429 )( 35993 )

Fifth iteration:

359932 - 17 = ( 25 )( 179 )( 226169 )

And I mused at that time about why it was going so slow. I did do two more iterations and you can see my musings back then in a post.

But it's interesting, I'm not sure why I thought that would work. And I'm also not so sure on why it should work. Though I'm guessing there is a simple reason. Putting it here to kind of collect it.

Maybe not the most practical thing for finding primes from what I'm seeing, but kind of curious. I think you can have cases where no larger prime comes up, but not sure.

Gives me something to do if I'm bored trying to understand something I used to know, years ago.

It does fade away. Sometimes I struggle to understand my own postings, with enough distance I am not quite certain how I figured something out.

But back then it was so easy I often zoomed through the results, not always leaving enough detail even for myself later on. I try not to do that now.

Have gained perspective.

James Harris

Much better than expected

There is that wonderful thing that time can do which is settle things. And as things have settled I've become far more appreciative of how well things have gone for me, which requires focus on the mathematical community.

Turns out it's very vibrant, and quite healthy, driving a TREMENDOUS amount of progress globally, and in the past I was worried about that community as I lacked perspective, and didn't realize that I'm a member of it.

Reality is that most mathematics used today is in science and technology, but also in finance, and vast other areas. And if you look at drivers for progress in mathematics itself, the real heat is in such areas.

It may not be admitted but the dying areas are "pure". And while there is a lot of funding for those areas which may dominate at universities, my experiences which I've pointed out here more recently explain why it is dying. Quite simply some have chosen to see mathematics as a social enterprise where the opinion of mathematicians is what's right.

Linking to an article from 2003 which I'm glad is still up. I've wondered why it was never pulled as it puts forward a notion that mathematics is too complex for absolute proof, so you need a committee of expert mathematicians, to decide.

My reaction was to define proof, where I focused on beginning with a truth and proceeding by logical steps so that you know it is absolute. And I recently put up an example of an absolute proof to refresh.

That attitude that the opinions of mathematicians are real in mathematics is so ludicrous it could never hold sway outside of "pure mathematics" which is useless, as in the real world buildings would literally crumble, as mathematics is not about opinion.

However I DO have a responsibility for the health of pure mathematics. And reality is that much of what I've accomplished is pure. But there is no rush. The world isn't pressuring me to do anything spectacular in that regard, which is a relief.

The ability of some to block mathematical proof, like with my stunning story of a paper revealing my ability to put forward a mathematically correct paper by established standards which appears to establish something not true is all about the power of uselessness.

The fight for the soul of mathematics could not be an easy one. But my relief was in knowing that the mathematical community is hale and hearty, powering our world forward. Which gives me time to just grind out the rest.

At times I've debated with myself if I should just force the situation, but use of force is not in my nature. While I might simply break the resistance of the pure mathematicians in an instant, the fallout could be problematic for decades.

Besides, the world does not need that to happen or it would pressure me to make it happen.

And so far the world seems ok with things going as they have.

Time can be a wonderful way for all concerned to gain perspective, especially me.

James Harris

Tuesday, August 04, 2015

Quadratic residue pairs and urge to know why

One of my favorite results explains the count of things called quadratic residue pairs. And while the way to count for primes has been known, I found a way to derive the result previously deduced. And it took some modular arithmetic and an alternate form to a well-known equation.

For example the quadratic residues of 31 are:

1, 2, 4, 5, 7, 8, 9, 10, 14, 16, 18, 19, 20, 25, 28

And there are 7 pairs: {1,2}, {4,5}, {7,8}, {8,9}, {9,10}, {18,19}, {19, 20}

Why would people care? Curiosity. And it's logical.

For a prime p, the quadratic residue pair count is [(p-1)/4], if p mod 4 = -1, or

(p-5)/4 if p mod 4 = 1.

Here I'm using [] for the floor() function, where for example [10/6] = 1, so just means throw away the remainder.

So with 31, since 31 mod 4 = -1, you have [(31 - 1)/4] = 7

That has been known for some time, but was figured out in some boring way that doesn't interest me as I found my own way!

I use (n-m)2 - Dm2 = 1, which is just an alternate form of x2 - Dy2 = 1. And consider it modulo D-1 to get:

2m = n-1(n2 - 1) mod D-1

(n - m)2 = m2 + 1 mod D-1

That second expression lets you see the quadratic residue pairs being forced! For people who are into numbers it's like this profound thing. Um, I guess it is. Should I try to speak for everyone interested in numbers? Well, for me it was just stunning. It still is. I just like to stare at it.

Notice there is no mention of primeness there as D-1 doesn't have to be prime. So it turns out there may not be solutions for m and n modulo D-1, because there are situations where there are no quadratic residues pairs. The math has to handle those situations of course.

I just LOVE that thing, so came up with a name for it.

Call it a quadratic residue engine.

Coming up with names is an awesome perk of discovery, but it's not intuitive! You need to think long and hard on it to get a style. It really bounces things along a bit. People like it more when you name these things.

Notice it doesn't care if D-1 is prime either, but that makes it easier to solve for the count, while you can use these equations to count in general, which I discussed as a possibility and is an example of something I can do later if get curious enough.

Go over far more including full derivation of count for primes in a post:

What makes this result more interesting is that it is the first time the count is derived versus being deduced. Go out there and find the result from others to see how that was done. Think it was some kind of pigeonhole thing or something.

But I just figure it out directly. To me that's a LOT more satisfying as you can even in this instance literally see the mechanism forcing the existence of quadratic residue pairs and then work out the logic for how this mathematical machinery lets you count them.

Not surprisingly to me, it's one of my most popular results based on search dominance.

It tells you the 'why' and satisfies the urge to know with numbers.

James Harris

Monday, August 03, 2015

My math paper and recent discussion here

Finally got a bit of discussion on this blog! And was on my post with some important facts and observations, where I think the two people commenting expressed things I should address in a post.

It has been well over a decade since I had a paper published in a formally peer reviewed mathematical journal, and maybe shouldn't be surprised if people end up speculating around the details. The paper was pulled from the electronic journal directly, where I guess they just tried to delete out the content. The table of contents were changed to say the paper was withdrawn. I did not withdraw the paper. The editors tried.

But EMIS kept the paper up, as well as the rest of the journal, as it managed one more edition and then ended operations. I like to say, it keeled over and died.

Am going to ask some questions on behalf of others based on what the commenters said. I'm guessing, so if you have other questions feel free to give them in the comments!

Ok, so some may wonder, did the editors know I was not a mathematician?

Yes. I told them at the outset when I submitted the paper. It took nine months for them to publish and was in communication with the editors routinely to see how it was going, and also wondering if they would just tell me no at some point. Was surprised and gratified when they notified me they would publish.

Is it possible the editors were not aware of the implications of the paper?

Can't say for sure, but I doubt it. The point of the paper was to show an important result, which is related to a coverage problem, but also to reveal that the currently established methods of rigor could be breached. As the paper by the rules appears to prove a result true in the ring of algebraic integers, which can be shown to NOT be true in that ring. It may be subtle to non-mathematicians but these were mathematical experts who probably understood it better than I did!

Did Usenet play a significant role as some posters seem to think?

Maybe but not the one they think they did, I'm sure. The reality is that Usenet is a fringe area, where people spend a lot of time making bold claims and well, insulting each other. My own view is that the editors expected a reaction from the global mathematical community of their peers, which did not occur. Then the Usenet posters emailing them would have been insult to injury as they say.

What if the editors had backed the paper do I think much would have changed?

In retrospect, I don't. My own guess is that the situation could be much as it is to this day! Only thing is I'd have some established mathematicians along for the ride. Maybe we would get together every once in a while and have some beers grousing over the situation. And possibly those mathematicians might have destroyed their careers, if they didn't anyway. But for me it's not a big deal in the same way as I'm NOT a mathematician.

Is a big deal in ways, of course. But not a career impacting big deal.

I say the situation is very political. And in fact I think mathematicians moved away from rigor especially with journals to human judgment and the view that mathematicians ultimately decide what is mathematically correct or not.

And I responded eventually by putting up a definition of mathematical proof on this blog, which removed any doubt about mathematical correctness being an absolute and NOT about shades of gray or opinion of mathematical experts.

My current view is that a political situation in this area will play itself out over time, am gratified that EMIS kept my paper up. Feel a bit sorry for mathematicians pulled into this drama. And am not terribly surprised that my paper wasn't simply accepted by the mathematical establishment with its implications.

Of course I'm a really big deal as a result of it I'm sure. I feel confident every major mathematician on the planet knows who I am.

To accept it mathematicians would have to shift from some of their most deeply held views and accept that like other disciplines, like my favorite which is physics of course, they can be forced to adjust.

Oh yeah, and feedback DOES matter! It helps me a lot to see what people may find puzzling or where they need clarification on things. Otherwise I'm just putting up what I think might be useful to people who have an interest in these things.

James Harris