### Friday, October 08, 2010

## JSH: Failure. Failure. Failure

Accepting that things are not going well. I think it's past time to abandon Usenet again. And I'll shift away from mathematics to other things.

Usenet is a bust. I think I was hoping for something else and convinced myself it was more, but I'll agree with the consensus opinion that it's now a dead medium.

The world has moved on. With that said I wouldn't be surprised if I end up posting again if things just get worse on other angles and sitting in muck is still all I have left.

You have your swamp back. Your dead end zone.

Usenet is dead for real work.

Usenet is a bust. I think I was hoping for something else and convinced myself it was more, but I'll agree with the consensus opinion that it's now a dead medium.

The world has moved on. With that said I wouldn't be surprised if I end up posting again if things just get worse on other angles and sitting in muck is still all I have left.

You have your swamp back. Your dead end zone.

Usenet is dead for real work.

### Sunday, October 03, 2010

## JSH: Usenet is a stage

One of the weirder things over the years I've seen are replies from posters who will talk about sci.math as if it's about a dozen posters, act as if I'm talking to them exclusively, and confidently tell me I should just quit bothering

Most people find public speaking to be terrifying. And for most their world IS made up of roughly a dozen or so people they communicate with on a regular basis.

Some posters appear to just transfer that local feel to Usenet. Which probably allows them to operate here.

If they see Usenet for what it is maybe they'd be terrified of the public speaking to such a huge audience.

But Usenet is global. Potentially you can talk to millions of people if you say something interesting enough, though reality is usually in the few hundreds, while some people like me—according to Google stats again—appear to routinely be read by thousands.

THOUSANDS. Not a few dozen!!!!!!!!

Years ago in reply to one of these people I actually asked them how many people they thought were reading my threads or something of that nature as I'm vague on the specifics of the question but their answer has stuck with me through the years: about 12.

To such people Usenet is a local thing. I'm bugging them in their hangout, and why won't I just leave?

For me Usenet is a global stage from which I can put forward my math ideas and there aren't many such places.

Not in the mainstream—Usenet is THE place for someone like me to put forward mathematical ideas, and it's hard to see why I'd go anywhere else, until I crossed over into the mainstream, and then I wouldn't need Usenet.

To quantify the impact, once I bought ads for my math blog for a while, and quickly spent several hundred dollars with ads in a few countries around the world managing to do that in about two weeks. Meanwhile my blog had received that many hits and more on its own!!!

But the money was well spent as I could better quantify the value of the organic hits to my blog by comparing with what it would cost to drive traffic using ads and that was really informative.

I'd guess that Usenet for me is roughly worth about $10,000 US per month on a conservative estimate in terms of ad spend if I were pushing the information out directly without it.

So I probably get about $100k US per year benefit from using Usenet. One hundred thousand dollars U.S. for those in countries who present dollar amounts differently than here in the United States.

And to posters who think that a nasty reply, calling me names, or endlessly claiming my research is worthless will make a difference against $100k per year, they're not living in the real world.

They don't have a snowballs chance in hell.

**them**as "no one" is listening to me!!!!!Most people find public speaking to be terrifying. And for most their world IS made up of roughly a dozen or so people they communicate with on a regular basis.

Some posters appear to just transfer that local feel to Usenet. Which probably allows them to operate here.

If they see Usenet for what it is maybe they'd be terrified of the public speaking to such a huge audience.

But Usenet is global. Potentially you can talk to millions of people if you say something interesting enough, though reality is usually in the few hundreds, while some people like me—according to Google stats again—appear to routinely be read by thousands.

THOUSANDS. Not a few dozen!!!!!!!!

Years ago in reply to one of these people I actually asked them how many people they thought were reading my threads or something of that nature as I'm vague on the specifics of the question but their answer has stuck with me through the years: about 12.

To such people Usenet is a local thing. I'm bugging them in their hangout, and why won't I just leave?

For me Usenet is a global stage from which I can put forward my math ideas and there aren't many such places.

Not in the mainstream—Usenet is THE place for someone like me to put forward mathematical ideas, and it's hard to see why I'd go anywhere else, until I crossed over into the mainstream, and then I wouldn't need Usenet.

To quantify the impact, once I bought ads for my math blog for a while, and quickly spent several hundred dollars with ads in a few countries around the world managing to do that in about two weeks. Meanwhile my blog had received that many hits and more on its own!!!

But the money was well spent as I could better quantify the value of the organic hits to my blog by comparing with what it would cost to drive traffic using ads and that was really informative.

I'd guess that Usenet for me is roughly worth about $10,000 US per month on a conservative estimate in terms of ad spend if I were pushing the information out directly without it.

So I probably get about $100k US per year benefit from using Usenet. One hundred thousand dollars U.S. for those in countries who present dollar amounts differently than here in the United States.

And to posters who think that a nasty reply, calling me names, or endlessly claiming my research is worthless will make a difference against $100k per year, they're not living in the real world.

They don't have a snowballs chance in hell.

### Saturday, October 02, 2010

## JSH: Counting primes as a puzzle

I've realized that giving people something they can intellectually attack on their own is maybe preferable, when feasible, than just tossing a solution at them, so I'm trying to present what I think is a fascinating puzzle counting prime numbers, where maybe you need a little history to get going.

A guy named Legendre is credited with a simple approach to counting primes, as you use a negative, since

Rather than write floor() all the time, another more concise usage is[], so [100/3] = 33, and floor just means to chop off the remainder.

So to count primes you do the opposite! You count composites and subtract that count. So continuing on, 100/5 = 20—didn't need floor() there—and [100/7] = 14. Adding them up: 50 +33+20+14 = 117, which is greater than 100, so what's wrong?

Well, there are 33 numbers divisible by 3, but 3 itself is divisible by 3 so you're counting it, and also, there are evens divisible by 3, like 6, so you double-counted.

Legendre was a smart guy so he fixed both those problems, but he did it in a highly particular way. For 33, he now counted how many were even! So you have [33/] = 16, and subtracting gives 33-16 = 17, and subtract 1 for 3, so you have 16 odd composites divisible by 3, up to 100.

For 5 it's even worse!!! You subtract [20/3] and 20/2, but now you have

(The actual composites are 25, 35, 55, 65, 85 & 95.)

Can you find a better way at each prime so that for instance at 5, you'd only count 6 composites, without even bothering to count the evens and numbers divisible by 3 that are also divisible by 5?

Once you have your composite count you just subtract from the total, like for 100, there are 74 composites, and 1 itself, so you have 100 - 75 = 25 primes.

Legendre's idea is usually called Legendre's Method and you can get the full thing easily enough on the web. The task here is to improve on the basic idea in a particular way, which is to only count composites at each prime that have NOT already been counted!

Awesomely in this instance that simple requirement can lead to a complete mathematical function.

I can tell you that the answer I got is a multi-dimensional P(x,y) function that uses a partial difference equation.

If you don't know what a partial difference equation is, it's the discrete analog of a partial differential equation. That is, it's just a discrete partial differential equation.

Ok then, puzzle away. If you give up, just Google: my prime counting function

Yup. I'm serious!!! You do need the "my" on the front as well to get mine.

A guy named Legendre is credited with a simple approach to counting primes, as you use a negative, since

**composites**are easy to count. Like up to 100, all the evens are 100/2, or 50 evens, while there are floor(100/3) = 33 numbers divisible by 3. (Say that three times fast!)Rather than write floor() all the time, another more concise usage is[], so [100/3] = 33, and floor just means to chop off the remainder.

So to count primes you do the opposite! You count composites and subtract that count. So continuing on, 100/5 = 20—didn't need floor() there—and [100/7] = 14. Adding them up: 50 +33+20+14 = 117, which is greater than 100, so what's wrong?

Well, there are 33 numbers divisible by 3, but 3 itself is divisible by 3 so you're counting it, and also, there are evens divisible by 3, like 6, so you double-counted.

Legendre was a smart guy so he fixed both those problems, but he did it in a highly particular way. For 33, he now counted how many were even! So you have [33/] = 16, and subtracting gives 33-16 = 17, and subtract 1 for 3, so you have 16 odd composites divisible by 3, up to 100.

For 5 it's even worse!!! You subtract [20/3] and 20/2, but now you have

**another**problem, as some of the numbers divisible by 3 are even! So you subtracted them twice!!! So Legendre added them back in with [20/6]. Yuck! You subtracted them out, and now you're adding them back! Way inefficient. Don't you feel silly now?(The actual composites are 25, 35, 55, 65, 85 & 95.)

Can you find a better way at each prime so that for instance at 5, you'd only count 6 composites, without even bothering to count the evens and numbers divisible by 3 that are also divisible by 5?

Once you have your composite count you just subtract from the total, like for 100, there are 74 composites, and 1 itself, so you have 100 - 75 = 25 primes.

Legendre's idea is usually called Legendre's Method and you can get the full thing easily enough on the web. The task here is to improve on the basic idea in a particular way, which is to only count composites at each prime that have NOT already been counted!

Awesomely in this instance that simple requirement can lead to a complete mathematical function.

I can tell you that the answer I got is a multi-dimensional P(x,y) function that uses a partial difference equation.

If you don't know what a partial difference equation is, it's the discrete analog of a partial differential equation. That is, it's just a discrete partial differential equation.

Ok then, puzzle away. If you give up, just Google: my prime counting function

Yup. I'm serious!!! You do need the "my" on the front as well to get mine.

### Friday, October 01, 2010

## JSH: More on using Usenet, your Usenet persona

Usenet is a pure attention zone, but like any other attention zone in our modern world it requires thought and effort on how you proceed and yes, I've paid a lot of attention to the US entertainment industry, as that's what I grew up with, and it has a lot of language that is familiar to me, and one thing you need at some point is a persona.

So it's a role you play. You get out here and you post and you're posting from a role. It's not you. It's the role that works for you.

And the thing about that is it isn't something I think you can really have explained so much, or that there is a method that I know, though others may, I guess, but it is critical so I've kind of left it as one of the latter things I'd talk about when it comes to using Usenet, as you have to get to a place where you're saying, that's not me, that's a role.

But why?

Well for one thing you may have people ripping on you day and night. Insulting you every way they can imagine, but more importantly people aren't really interested in you.

There are about 6.8 billion people on this planet, who are you?

Your ideas are the driver, as I think I've shown, but a person has to be behind those ideas, but the person shouldn't get in their way.

Your ideas are the engines. If you put yourself out there then you may defend yourself at some point to the detriment of your ideas which are the only things that can pull you forward.

So your persona protects YOU.

Your ideas pull you forward.

But if you take things personally then you may get in the way of your ideas.

So it is like the entertainment industry in that respect in that to some extent what you can do, maybe can be said to be acting, but not really as you're just typing things that can be read by thousands of people around the world.

It's ideas first. People are secondary. Your goddamn personality is not important here—your persona is to some extent. Who you are deep inside is not relevant.

It is your ideas. Your math.

Nothing else ultimately will take you anywhere with Usenet as it is a pure attention zone and if your math is not up to the challenge then nothing else is going to do it.

But just don't get in the way of your ideas.

So it's a role you play. You get out here and you post and you're posting from a role. It's not you. It's the role that works for you.

And the thing about that is it isn't something I think you can really have explained so much, or that there is a method that I know, though others may, I guess, but it is critical so I've kind of left it as one of the latter things I'd talk about when it comes to using Usenet, as you have to get to a place where you're saying, that's not me, that's a role.

But why?

Well for one thing you may have people ripping on you day and night. Insulting you every way they can imagine, but more importantly people aren't really interested in you.

There are about 6.8 billion people on this planet, who are you?

Your ideas are the driver, as I think I've shown, but a person has to be behind those ideas, but the person shouldn't get in their way.

Your ideas are the engines. If you put yourself out there then you may defend yourself at some point to the detriment of your ideas which are the only things that can pull you forward.

So your persona protects YOU.

Your ideas pull you forward.

But if you take things personally then you may get in the way of your ideas.

So it is like the entertainment industry in that respect in that to some extent what you can do, maybe can be said to be acting, but not really as you're just typing things that can be read by thousands of people around the world.

It's ideas first. People are secondary. Your goddamn personality is not important here—your persona is to some extent. Who you are deep inside is not relevant.

It is your ideas. Your math.

Nothing else ultimately will take you anywhere with Usenet as it is a pure attention zone and if your math is not up to the challenge then nothing else is going to do it.

But just don't get in the way of your ideas.

## JSH: Questioning world appeal

I keep mentioning country counts as they really have me wondering and I don't have the full answers yet, but I can't just dismiss them as I've been looking at web analytics data now for years, and I know that you just don't have your work appeal to people in roughly 40 countries month after month, easily. It's just kind of hard to do, and the thing about it is, I'm not appealing. It's my math.

So my math is this driver that doesn't really bring me along, so while I can' get the definition of mathematical proof with a search on—definition of mathematical proof—a search on my name in Google gets, Crank.net and it's not even in the top 10!!! So my personal negatives are HUGE.

People aren't clicking on my math blog from 120 countries for anything about me. It's my math.

Oh, and it is a harsh negative to Usenet that you can have these hanger's on who will reply to you like they're big important people—aka BIP's—and they say the stupidest crap. So they will say, with all seriousness—that Google search results are meaningless.

Yeah, for them, as they don't have dominance in the search result arena with anything important, and I've learned one crucial thing: people lie their asses off.

One thing I've learned more and more over the years is that people will just lie to you. And the bigger the thing, the more you'll find people who will line up to lie.

So ran the numbers of the year for my math blog, and only up to 98 countries/territories this year according to Google Analytics.

Thousands of cities and you wonder—what were those people thinking? Oh, so how many cities? Typed that about "thousands" so got curious and actually checked—imagine a pause as I went to see—it's only 1974 cities. So not quite thousands.

I'm in the US and I don't know any blogs I read even from the UK, let alone, say, Stockholm, which was the #5 city (London is as usual #1 for visits). If it's not in English I wouldn't be able to do anything with it anyway.

I can look at stats across my blogs for a lot of languages and wonder about that as well.

And some freaking nobodies who have ripped on me for YEARS from back when none of that was true will tell me today that Google hit counts mean nothing? That Google search results mean nothing? And talk as if anyone with a website gets hits from just about the entire world?

People will lie to you about just about anything.

People lie.

These turds sit in my threads. I can't put up a thread without one of these idiots replying.

And they reply just like they did years ago! They're frozen in freaking time, but they clog up the thread. Reply just about as soon as I put something up, and I rarely hear from anyone else.

I used to call them attention parasites. I think of them as attention barnacles. They settle on the big ship, and increase the drag, but how do these people take themselves seriously? The angry idiots. That's another name I've called them at times.

What do they see when they look in the mirror?

I think they see someone that few people notice. I think they see a person who mostly is ignored in the big wide world.

So they rush to Usenet to see if I've put up a thread and they can post a reply in it…

So my math is this driver that doesn't really bring me along, so while I can' get the definition of mathematical proof with a search on—definition of mathematical proof—a search on my name in Google gets, Crank.net and it's not even in the top 10!!! So my personal negatives are HUGE.

People aren't clicking on my math blog from 120 countries for anything about me. It's my math.

Oh, and it is a harsh negative to Usenet that you can have these hanger's on who will reply to you like they're big important people—aka BIP's—and they say the stupidest crap. So they will say, with all seriousness—that Google search results are meaningless.

Yeah, for them, as they don't have dominance in the search result arena with anything important, and I've learned one crucial thing: people lie their asses off.

One thing I've learned more and more over the years is that people will just lie to you. And the bigger the thing, the more you'll find people who will line up to lie.

So ran the numbers of the year for my math blog, and only up to 98 countries/territories this year according to Google Analytics.

Thousands of cities and you wonder—what were those people thinking? Oh, so how many cities? Typed that about "thousands" so got curious and actually checked—imagine a pause as I went to see—it's only 1974 cities. So not quite thousands.

I'm in the US and I don't know any blogs I read even from the UK, let alone, say, Stockholm, which was the #5 city (London is as usual #1 for visits). If it's not in English I wouldn't be able to do anything with it anyway.

I can look at stats across my blogs for a lot of languages and wonder about that as well.

And some freaking nobodies who have ripped on me for YEARS from back when none of that was true will tell me today that Google hit counts mean nothing? That Google search results mean nothing? And talk as if anyone with a website gets hits from just about the entire world?

People will lie to you about just about anything.

People lie.

These turds sit in my threads. I can't put up a thread without one of these idiots replying.

And they reply just like they did years ago! They're frozen in freaking time, but they clog up the thread. Reply just about as soon as I put something up, and I rarely hear from anyone else.

I used to call them attention parasites. I think of them as attention barnacles. They settle on the big ship, and increase the drag, but how do these people take themselves seriously? The angry idiots. That's another name I've called them at times.

What do they see when they look in the mirror?

I think they see someone that few people notice. I think they see a person who mostly is ignored in the big wide world.

So they rush to Usenet to see if I've put up a thread and they can post a reply in it…

## JSH: What more?

I see replies to my posts from the SAME PEOPLE ALL THE TIME and they don't say anything new or interesting to me. But I've finally accepted that I get a big boost from Usenet, as I can look at analytics data across my websites—3 blogs plus—where primary impact is on my math blog. I don't like telling a lot about the data, but simply presenting the "core error" as a puzzle gave a massive boost to my blog, and I will say that I've now had hits from 50 countries in the last 30 days. A nice jump.

And it was FAST. So I'm thinking more about Usenet now.

But I'm concerned about efficiency. Certain posters have obsessively replied to me for years and they're stuck in a time warp.

For them I'll always be just this little crank that they put down and dismiss, who refuses to kiss their ass, but who desperately—they think—just wants their approval. But their replies I think drive out others, so I'm not seeing the maximum use of Usenet that I'd like.

So for OTHERS, what kinds of things do you need to know about my math? What more? As in more of what kind of information?

More of the research as puzzles? Maybe even—God help me because I hate the thought—homework?

As I shift from insult based antagonism, is that a bad idea? I think that conflict is a driver but I'm wary as the world grows up online of being too nasty in replies to people. But is softening just bad for the entertainment aspect?

How much entertainment is actually needed?

Are there any areas of focus: prime residues more? prime gap more? solving residues more? "core error" more? Diophantine equations more? Logic more?

Over a decade I've managed to cover just about all of number theory and a good chunk of logic. And I like poking at established researchers as I can take over search results. Kind of get a kick out of the thought of those guys—mostly guys you know—furious whenever they do web searches seeing my math pop up!

Oh, so maybe that could be fun?

What areas of mathematics might I take over next? What established researcher might I poke by pushing my own research up ahead of his in web searches?

At this point I have a HUGE ability to go into just about any area of mathematics at will. And it doesn't take much effort.

And it was FAST. So I'm thinking more about Usenet now.

But I'm concerned about efficiency. Certain posters have obsessively replied to me for years and they're stuck in a time warp.

For them I'll always be just this little crank that they put down and dismiss, who refuses to kiss their ass, but who desperately—they think—just wants their approval. But their replies I think drive out others, so I'm not seeing the maximum use of Usenet that I'd like.

So for OTHERS, what kinds of things do you need to know about my math? What more? As in more of what kind of information?

More of the research as puzzles? Maybe even—God help me because I hate the thought—homework?

As I shift from insult based antagonism, is that a bad idea? I think that conflict is a driver but I'm wary as the world grows up online of being too nasty in replies to people. But is softening just bad for the entertainment aspect?

How much entertainment is actually needed?

Are there any areas of focus: prime residues more? prime gap more? solving residues more? "core error" more? Diophantine equations more? Logic more?

Over a decade I've managed to cover just about all of number theory and a good chunk of logic. And I like poking at established researchers as I can take over search results. Kind of get a kick out of the thought of those guys—mostly guys you know—furious whenever they do web searches seeing my math pop up!

Oh, so maybe that could be fun?

What areas of mathematics might I take over next? What established researcher might I poke by pushing my own research up ahead of his in web searches?

At this point I have a HUGE ability to go into just about any area of mathematics at will. And it doesn't take much effort.

### Thursday, September 30, 2010

## JSH: Puzzling counting prime numbers

Rather than give directly a mathematical find I've found it interesting to see if others can figure out something on their own, thus giving them an intellectual challenge. In this thread the challenge is to find a certain way to count prime numbers.

There are LOTS of resources available with a lot of mathematical literature on the subject, and all of the established literature is available for this challenge. Of course I also have the answer, which is not in established literature, for those who give up.

Challenge:

Find a prime counting function P that counts primes by first counting composites and subtracting that count and 1 from the total count to get primes, which only counts composites at each prime that have not already been counted.

That is, Legendre's method rather inefficiently counts ALL composites at each prime that have that prime as a factor, which is dumb!!! It's so dumb you end up just subtracting those back out which is a naive way to count, don't you think? Why, for instance, if you're counting composites divisible by 3, should you also count evens? Of if you're counting composites with 5 as a factor, count the ones with 3 and 2, and then just subtract them back out? It's WACKY!

Can you find a more efficient prime counting method than Legendre's which more smartly counts ONLY those composites at each prime that have not ALREADY been counted?

If you think the problem is too easy, great! Give the function in reply.

If you have no clue, here's a clue.

Hint: The requirements force a P(x,y) function, so you end up with a multi-dimensional prime counting function instead of a pi(x), single variable one.

Can anyone solve the puzzle on their own? If you get completely lost, of course, the answer is on my math blog. Given lots of places as I've explained, and explained and explained.

But if I can figure it out, can you?

There are LOTS of resources available with a lot of mathematical literature on the subject, and all of the established literature is available for this challenge. Of course I also have the answer, which is not in established literature, for those who give up.

Challenge:

Find a prime counting function P that counts primes by first counting composites and subtracting that count and 1 from the total count to get primes, which only counts composites at each prime that have not already been counted.

That is, Legendre's method rather inefficiently counts ALL composites at each prime that have that prime as a factor, which is dumb!!! It's so dumb you end up just subtracting those back out which is a naive way to count, don't you think? Why, for instance, if you're counting composites divisible by 3, should you also count evens? Of if you're counting composites with 5 as a factor, count the ones with 3 and 2, and then just subtract them back out? It's WACKY!

Can you find a more efficient prime counting method than Legendre's which more smartly counts ONLY those composites at each prime that have not ALREADY been counted?

If you think the problem is too easy, great! Give the function in reply.

If you have no clue, here's a clue.

Hint: The requirements force a P(x,y) function, so you end up with a multi-dimensional prime counting function instead of a pi(x), single variable one.

Can anyone solve the puzzle on their own? If you get completely lost, of course, the answer is on my math blog. Given lots of places as I've explained, and explained and explained.

But if I can figure it out, can you?

### Wednesday, September 29, 2010

## JSH: Puzzling through number theory?

With the success of the approach of presenting a major number theory issue as a puzzle I've began to consider that the problem with my prior approach has been that I've not given people the opportunity for intellectual challenge. Simply explaining mathematics may not be nearly as enticing as puzzling through.

So might that help in other areas? Conceivably here's another area for those who wish to match wits with me.

Find a prime counting function P that counts functions by first counting composites and subtracting that count and 1 from the total count to get primes, which only counts composites not already counted.

That is, Legendre's method rather inefficiently counts ALL composites at each prime that have that prime as a factor, which is dumb!!! Why, for instance, if you're counting composites divisible by 3, should you also count evens? It's WACKY!

Can you find a more efficient prime counting method than Legendre's which more smartly counts ONLY those composites at each prime that have not ALREADY been counted?

As a check, how hard is that as a puzzle? Should a smart, say, undergrad be able to succeed? How about a grad student?

If you think the problem is easy, great! Give the function in reply. If you have no clue, here's a clue.

Hint: The requirements force a P(x,y) function, so you end up with a multi-dimensional prime counting function instead of a pi(x), single variable one.

Can anyone solve the puzzle on their own? If you get completely lost, of course, the answer is on my math blog. Given lots of places as I've explained, and explained and explained.

But if I can figure it out, can you?

So might that help in other areas? Conceivably here's another area for those who wish to match wits with me.

Find a prime counting function P that counts functions by first counting composites and subtracting that count and 1 from the total count to get primes, which only counts composites not already counted.

That is, Legendre's method rather inefficiently counts ALL composites at each prime that have that prime as a factor, which is dumb!!! Why, for instance, if you're counting composites divisible by 3, should you also count evens? It's WACKY!

Can you find a more efficient prime counting method than Legendre's which more smartly counts ONLY those composites at each prime that have not ALREADY been counted?

As a check, how hard is that as a puzzle? Should a smart, say, undergrad be able to succeed? How about a grad student?

If you think the problem is easy, great! Give the function in reply. If you have no clue, here's a clue.

Hint: The requirements force a P(x,y) function, so you end up with a multi-dimensional prime counting function instead of a pi(x), single variable one.

Can anyone solve the puzzle on their own? If you get completely lost, of course, the answer is on my math blog. Given lots of places as I've explained, and explained and explained.

But if I can figure it out, can you?

### Monday, September 27, 2010

## JSH: Attacking the puzzle

Rather than do yet ANOTHER explanation of an issue in number theory, I presented the issue as a puzzle. Some posters have replied in that thread, and those who tried to solve the puzzle failed massively, but protested when I noted that they'd failed.

So here is an attack on the puzzle using the knee-jerk responses that are WRONG. So it's easily to massively fail against this thing.

Here again is the puzzle, and I'll then attack it just a bit.

Try in the ring of algebraic integers to use:

9(g_1(x) + 1)(g_2(x) + 2) = (f_1(x) + 9)(f_2(x) + 9) = 9*P(x)

where g_1(0) = g_2(0) = 0, and one of the f's equals 0 at x=0, as well, while P(x) is a quadratic with integer coefficients.

You can find cases in the ring of algebraic integers where NEITHER of the f's can have 9 as a factor, for certain values of x, so you wander off the to the complex plane—because you've a VERY SMART math student—so you can see exactly what is happening.

There you note that:

(9g_1(x) + 9)(g_2(x) + 2) = (f_1(x) + 9)(f_2(x) + 9) = 9*P(x)

follows trivially by the distributive property so FOR THAT CASE the ring of algebraic integers presumably would have 9 as a factor for all x, but can be shown to NOT have 9 as a factor as noted above.

Being an EXTREMELY INTELLIGENT math student you find this puzzling, so you consider it until it makes sense.

First off, let's attack the idea of going to the complex plane as a divisibility issue as it DOES NOT MATTER on the complex plane.

Well, note that the result HOLDS for polynomial functions for the f's and g's. So with all integers—valid on the complex plane—you do indeed find that one of the f's has 9 as a factor for all x.

Notice that following that on the complex plane does not remove the result.

The complex plane INCLUDES the ring of integers, so it can look at something happening in that ring easily.

If divisibility were the issue, it would emerge with integers as well, and everything would be divisible by 9. But one of the f's is not in general divisible by 9, so no, divisibility as the issue is a massive fail for this puzzle.

One thing that I think is fascinating as a bizarre issue is that some posters seem lost on the concept that you can use the complex field to simply look at something happening in a ring, like the ring of integers. May seem like overkill, but you can do it!

So making that mistake actually also says something about your sophistication as a math student, as presumably a fairly astute math student would realize that trivially.

Another poster actually said, there was no puzzle!!! Clearly that one is a massive fail as my easy counter is to explain it then. But his answer was to try and claim divisibility, so another massive fail. Claiming there is no puzzle but not being able to answer correctly is not an answer.

Can you now figure out the puzzle?

What others issue need I address? This thread can be for attacking the puzzle, the validity of it as a puzzle, or if you want, try to give an answer. I assure you it's not a trivial task to answer it correctly.

So here is an attack on the puzzle using the knee-jerk responses that are WRONG. So it's easily to massively fail against this thing.

Here again is the puzzle, and I'll then attack it just a bit.

Try in the ring of algebraic integers to use:

9(g_1(x) + 1)(g_2(x) + 2) = (f_1(x) + 9)(f_2(x) + 9) = 9*P(x)

where g_1(0) = g_2(0) = 0, and one of the f's equals 0 at x=0, as well, while P(x) is a quadratic with integer coefficients.

You can find cases in the ring of algebraic integers where NEITHER of the f's can have 9 as a factor, for certain values of x, so you wander off the to the complex plane—because you've a VERY SMART math student—so you can see exactly what is happening.

There you note that:

(9g_1(x) + 9)(g_2(x) + 2) = (f_1(x) + 9)(f_2(x) + 9) = 9*P(x)

follows trivially by the distributive property so FOR THAT CASE the ring of algebraic integers presumably would have 9 as a factor for all x, but can be shown to NOT have 9 as a factor as noted above.

Being an EXTREMELY INTELLIGENT math student you find this puzzling, so you consider it until it makes sense.

First off, let's attack the idea of going to the complex plane as a divisibility issue as it DOES NOT MATTER on the complex plane.

Well, note that the result HOLDS for polynomial functions for the f's and g's. So with all integers—valid on the complex plane—you do indeed find that one of the f's has 9 as a factor for all x.

Notice that following that on the complex plane does not remove the result.

The complex plane INCLUDES the ring of integers, so it can look at something happening in that ring easily.

If divisibility were the issue, it would emerge with integers as well, and everything would be divisible by 9. But one of the f's is not in general divisible by 9, so no, divisibility as the issue is a massive fail for this puzzle.

One thing that I think is fascinating as a bizarre issue is that some posters seem lost on the concept that you can use the complex field to simply look at something happening in a ring, like the ring of integers. May seem like overkill, but you can do it!

So making that mistake actually also says something about your sophistication as a math student, as presumably a fairly astute math student would realize that trivially.

Another poster actually said, there was no puzzle!!! Clearly that one is a massive fail as my easy counter is to explain it then. But his answer was to try and claim divisibility, so another massive fail. Claiming there is no puzzle but not being able to answer correctly is not an answer.

Can you now figure out the puzzle?

What others issue need I address? This thread can be for attacking the puzzle, the validity of it as a puzzle, or if you want, try to give an answer. I assure you it's not a trivial task to answer it correctly.

## JSH: Better as a puzzle?

I'm kind of excited at the idea of presenting what I claim is a major number theory issue as a puzzle.

And it kind of makes sense to do it that way versus trying to explain it in detail—which I've been doing for years—as it could be more fun that way for those who wish a massive intellectual challenge. After all, the problem has stood for over a hundred years!

Back-story as well is that I faced it as a puzzle years ago. As back in 2002 I had what I thought was a proof of Fermat's Last Theorem using equations that followed from what I call tautological spaces. YEARS of effort but I kept being called on the ring. Posters would bug me about the ring, and for a while I'd even talked about a "flat ring" as I needed a ring without fractions.

Well one poster says that what I really wanted was the ring of algebraic integers, and I said, ok, as they sounded fine, and went on about my way, except with this latest arguments posters noted my result contradicted with the ring of algebraic integers.

I was floored! It was like, uh oh, not good. And I began to puzzle it out. After some time I figured out the puzzle!

And I discovered the object ring.

It IS a puzzle.

Presumably the smartest math people can just figure it out! Which can keep me from having to keep trying to explain, but a major benefit is emerging already in the thread I created presenting the puzzle as some very obsessive posters just fail outright, and badly.

I'm re-thinking how I look at various posters based on responses in that thread. Surprisingly to me, I may have given some of them too much benefit of the doubt as to their, um, mathematical abilities.

That puzzle is a breaker. It will break anyone but the best. And responses to it, are crystal clear as to basic math ability.

Either you have it, or you don't.

And it kind of makes sense to do it that way versus trying to explain it in detail—which I've been doing for years—as it could be more fun that way for those who wish a massive intellectual challenge. After all, the problem has stood for over a hundred years!

Back-story as well is that I faced it as a puzzle years ago. As back in 2002 I had what I thought was a proof of Fermat's Last Theorem using equations that followed from what I call tautological spaces. YEARS of effort but I kept being called on the ring. Posters would bug me about the ring, and for a while I'd even talked about a "flat ring" as I needed a ring without fractions.

Well one poster says that what I really wanted was the ring of algebraic integers, and I said, ok, as they sounded fine, and went on about my way, except with this latest arguments posters noted my result contradicted with the ring of algebraic integers.

I was floored! It was like, uh oh, not good. And I began to puzzle it out. After some time I figured out the puzzle!

And I discovered the object ring.

It IS a puzzle.

Presumably the smartest math people can just figure it out! Which can keep me from having to keep trying to explain, but a major benefit is emerging already in the thread I created presenting the puzzle as some very obsessive posters just fail outright, and badly.

I'm re-thinking how I look at various posters based on responses in that thread. Surprisingly to me, I may have given some of them too much benefit of the doubt as to their, um, mathematical abilities.

That puzzle is a breaker. It will break anyone but the best. And responses to it, are crystal clear as to basic math ability.

Either you have it, or you don't.