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Monday, October 30, 2017

Knowledge gained to be shared

There is a benefit in our times with the web as enables wide sharing but you still need the people who pay attention for ideas to move, and deeply appreciate those who show interest in math content here. Some of it is very challenging in many ways, as have pushed some innovative concepts, much of which falls under the category like to call abstract reductionism.

To me have found great knowledge to be shared. And glad blog lets me to such an extent.

Have attempted to get comment on my research from people who are establishment in mathematics with far less ability to even get a response now. Possibly as a result of my past success. Also yes, know have been VERY critical of mathematicians, so maybe they're sensitive to that?

That is unfortunate. Means is rather worthless for me to push in that direction information on people who do not want it, and am wary of suggesting others share, but it is simply knowledge gained.

Have a math group where have ability to attach PDF's to posts: My Math Group

The point of it in the future is as a place where others can share rather than send me math ideas.

Now though my posts dominate and have highlighted some things, so thought should bring that up.

If any of you DO know mathematicians who will answer or are open to new ideas please feel free to share with them, or let me know and I will contact.

Oh yeah, kind of cool, but even contacted the NSA twice, with my modular inverse method. Gave it to them BOTH times as they have a form where could paste it in there as felt it my duty, you know, as a concerned citizen. They auto-responded first time, but not the second. Which I kind of think is hilarious. Have gotten through there at least once in the past with prior research.

Guess am too known now. Mathematicians it seems are not replying to me any more.

Maybe some of you can help?

Am not joking either. Looks like mathematicians have switched to full on silent treatment. At least in the past could get through to someone who would at least reply then do nothing.

The truth then needs others to transport where necessary. If you wish to help, please use your best judgement. In some situations could be dangerous for you.

The math includes areas related to national security of many nations. Some might appreciate the truth, while in others could be a bad idea to talk to them about at all.

So, if you decide to to help in that way, good luck.


James Harris

Sunday, October 29, 2017

When numbers interest

More recently have been looking more to what I call number authority, with demonstrations of number theory with actual numbers where is primarily sums of squares because that's easy for me with some of my research. Like:

349672 + 7522 + 1128+ 1880+ 26322 + 41362 + 48882 = 357212

_____________________________

Maybe was TOO easy for me. As didn't think much of my ability to do such things, until I did some searches to find how world's number theorists summed an arbitrary number of squares to get a square. And could not find evidence they knew how. Was so depressing for me, and disappointing too, will admit.

Disillusionment is not fun. And yes, have had plenty of reasons to realize before, but kept rationalizing. Finally realized there was no more room for benefit of the doubt.

Shifted my thinking and my approach to posts, such things. Is just one example. Maybe they do have a way. Web search used: sum squares to square

Checked again for this post. Maybe someone knows another? (Found something!!! I do keep searching and saw a paper from 1953, for a sum of 7 squares to a square.) For over a decade and before, wanted to think one way, and grudgingly have shifted to another as recent posts reflect. But like check this out.

Forgot I had this list:

1) 12 + 22 = 5

2) 32 + 42 = 25 = 52

3) 112 + 22 = 125 = 53

4) 72 + 242 = 625 = 54

5) 412 + 382 = 3125 = 55

6) 1172 + 442 = 15625 = 56

Which comes from a post from December 2014, when thought could do something clever to drum up interest.

People have to be interested in such things though. You cannot make them. And for a long time I rationalized that modern number theorists think such things trivial, and that they hearken back to older number theory which is well-worked and boring. But I've proven it is neither.

Recently came up with another list:

42 + 32 = 52

132 + 92 = 2(53)

72 + 242 = 54

792 + 32 = 2(55)

442 + 1172 = 56

3072 + 2492 = 2(57)

Why? Why not? I find numbers interesting and like to play with them. Also at times prefer that number authority that stands alone when I know there can't be a mistake, once I've checked enough to eliminate silly errors. The results are just perfect. They are absolute truths.

I find them comforting.

The social problem is not that big of a deal in the history of working mathematics. Science and technology were protected for the most part. Still, guess my job protecting the discipline of mathematics is slightly related. Have been more concerned with HUGE timeframes. Human beings come and go, but the numbers do not.

People who found a way to gain a success by pretending to be something that demonstrably does not interest them are also a problem I have to resolve, but so easily their work will be dismissed from human history. Just exclude anything presented with ring of algebraic integers as the ring should clear most of it, maybe all.

Even if I consider it dismissively, a social problem must I guess be handled.

Other mathematicians are apparently simply weak. Not smart enough to do the right thing either.

To me even the idea that it's even slightly worth it, shows how little they know of the value of perfect knowledge, or of the thrill of discovery.

The best? Accept nothing less than truth.

The best discoverers make history. Because truth works better. Duh.

Is WAY more fun too! Like, get to make cool pronouncements read all over the world.

Is SO cool.

Maybe I should take my responsibilities more seriously.

Thinking about it. Working on it.


James Harris

Simple Generalized Quadratic Factorization

Important in my research is a simple generalization of the factorization of a quadratic. That quadratic is easily considered in the complex plane.

In the complex plane:

P(x) = (g1(x) + 1)(g2(x) + 2)

where P(x) is a primitive quadratic with integer coefficients, g1(0) = g2(0) = 0, but g1(x) does not equal 0 for all x.

The simplest example is: P(x) = x2 + 3x + 2

Solving for g's in general is easily done with some substitutions, where one will seem superfluous, but is important. To solve with my approach we will need a new variable k, and two new functions.

Introduce k, where k is a nonzero, and new functions f1(x), and f2(x), where:

g1(x) = f1(x)/k and g2(x) = f2(x) + k-2

Multiply both sides by k, and substitute for the g's, which gives me the now symmetrical form:

k*P(x) =  (f1(x) + k)(f2(x) + k)

The purpose of forcing symmetry is to allow for solutions where also need one more function, then can solve for the f's using the quadratic formula.

And introduce H(x), where: f1(x) + f2(x) = H(x)

Then just solve for one of the f's, and substitute so I can find:

f12(x) - H(x)f1(x) - kH(x) - k2 + k*P(x) = 0

And then you can solve for f1(x) using the quadratic formula:

f1(x) = (H(x) +/- sqrt[(H(x) + 2k)2 - 4k*P(x)])/2

And H(x) is a handle for every possible factorization with the g's, and has a key constraint: H(0) = -k + 2.

If wished, as an exercise reader can confirm our simple factorization solution with:

k = 1, H(x) = 2x+1, and P(x) = x2 + 3x + 2

Which will give x and x+1 as solutions for the f's, and x as a solution for the g's. And, a curious reader might next consider what happens if you force a non-polynomial factorization, for instance with H(x) = 3x + 1. Notice you only need make sure H(0) = 1.

Have been considering in the complex plane, but notice can easily move to rings by how control variables. For example if k is an integer, and is unit, which means 1 or -1, and H(x) is an algebraic integer function, then the f's are forced to be as well. And also the g's are so forced.

So we know that we can get every possible algebraic integer factorization using our generalized approach, which is important.

However, we can also choose to pick some other k, like k = 2, or k = 3, Which is of mathematical interest beyond the scope of this instructional post.

Have introduced a generalized quadratic factorization with a simple quadratic, and stepped through a basic approach to finding solutions for that factorization in the complex plane.


James Harris

Thursday, October 26, 2017

Clarification on algebraic integer solution existence claims

Can be very difficult to escape error in a mathematical argument, and have realized that some earlier posts talking a generalized quadratic factorization were in error with regard to claims about existence of solutions in ring of algebraic integers. Have corrected more than one post and may have missed others, so will also have this post to clarify. Actually had corrected to current thinking by November 2015, but didn't realize needed to update prior posts until today. Guess just forgot was in them. It happens.

The generalized factorization is in complex plane:

P(x) = (g1(x) + 1)(g2(x) + 2)

where P(x) is a primitive quadratic with integer coefficients, g1(0) = g2(0) = 0, but g1(x) does not equal 0 for all x.

Introduced a mathematical approach which involves multiplying both sides by an integer k, which I choose to not be a unit in an important proof which is correct.

Introduce k, where k is a nonzero integer, and new functions f1(x), and f2(x), where:

g2(x) = f2(x) + k-2 and g1(x) = f1(x)/k,

multiply both sides by k, and substitute for the g's, which gives me the now symmetrical form:

k*P(x) =  (f1(x) + k)(f2(x) + k)

However, for a bit I forgot that detail of my choice with at least one post, which I've updated, but had further flawed thinking, and in the past claimed that the g's could not both be algebraic integers in general when non-rational. Which seemed so dramatic to me! And was wrong. Which again reminds me, emotion and math? Do not mix well.

Have now noted that they both are when k = 1 or -1, with approach for finding solutions for them that I discovered.

Is important enough should note have already posted full resolution explanation, which have had since 2015 but hadn't so elaborated, and will include pertinent information here for what happens if you try to force things with a different non-unit k using g's that are algebraic integers:

The algebra doesn't care, but will NOT let the f's be roots of the same quadratic with integer coefficients then with k not 1 or -1. They will disjoint from each other. What is key is the process where they are roots of the same quadratic with integer coefficients. As far as the math is concerned, you simply have one of the f's multiplied by k, so its removal to get one of the g's is as easy here as anywhere else.

So yeah, my thinking before was off, and actually discovered by looking at a case, and going, hey. Am SO lucky could find a proof with quadratics, as you can just put in some numbers and see an actual case. Then realized that yeah, with quadratics when you have something like 1+sqrt(3) and 1-sqrt(3), you need them to be the roots, for a quadratic with integer coefficients. Fiddle with one, and you disjoint the results, like add something to one, like 2, and have 3+sqrt(3) and 1-sqrt(3), and end up with roots at best of the same quartic with integer coefficients. Guess may as well fully explain how I noticed something actually kind of simple.

So realized it by doing the above. Looking at an actual case and then looking at the f's that resulted and seeing they couldn't be roots of the same quadratic with integer coefficients. Then it's like, duh.

My apologies for the error. If there are posts which still have it, please disregard. Or if you point them out will correct. And am still checking. Problem with something wild is that urge to plaster over a lot of posts, which have learned to try NOT to do.

Oh, and how did I notice finally had prior posts which were not updated to my current view? The usual, as was checking Blogger stats noted interest in a particular post. When looked at it to see why, realized mistake was there while reading through. Is a cool process. Not sure if was deliberate help, but in case it was? Thanks!!!

So yeah still rely on constant communication through the web, though now is primarily indirect. Am constantly getting feedback based on which posts draw interest, as well as from country stats as get an idea of who is paying attention. Is a daily thing. Now yeah, things are more, popular lately.

My earlier claim was that the algebra was blocking that form for algebraic integers, and that is in error.

This post is simply notice of that past mistake and that it has been corrected in current arguments.


James Harris

Friday, October 20, 2017

Information that draws attention

Great thing about using your own blog is can dismiss claims are simply seeking attention. Better when the math does that work--pulling interest.

Mathematics and attention are easy in that you can prove things, but can still be hard if feel need to convince. I prefer usually to just let the math do THAT work too. My job then? Explain well.

Web enables much.

Like, Blogger stats let me know there is global interest. In last 24 hours according to them have had visits from: United States, United Kingdom, Netherlands, Portugal and Ukraine

For the last 7 days though, Google Analytics has: United States, United Kingdom, Netherlands, Germany, Croatia, Italy, and Mongolia

Which is just for folks who the web analytics even show. Weekly list for Blogger actually maxes out at 10 countries, as they will not show more, but don't feel like typing them into here. Where yes, notice some mismatch between the lists! More like give a sense than certainty. Mainly shows global pull. Why the difference? Guess you'd have to ask Google. Google owns both Blogger and Google Analytics.

Main thing is, people from all over the world drawn to the mathematics presented on this blog. Better than trying to draw attention through other means. Turns out, only need to use my own things for a global reach as great as a major math journal.

Is remarkable though, as the math itself provides the draw, and regardless of other things reveals am a major discoverer. Makes sense! Am a person who experiences the power the mathematics has, in and of itself. The math can simply pull global attention.

With powerful mathematics, what happens then? I don't know. What I'd like to have happen is discussion. Pulling in of people to talk things over preferably copied off the blog.

Even better, write out a mathematical approach, for yourself, and discuss with others.

The math discovered is humanity's. And debates over absolute mathematical truth DO fade in time.

Human beings try to be logical. The math is logical.

Want some number authority? Then you might check this post: Some number examples

Has things like: 42 + 6+ 10+ 14862 = 882

Looking for a simple but cool pattern with exponents? Sum two squares and 5

Has things like: 792 + 32 = 2(55) and 442 + 1172 = 56

Need a quick overview of modular arithmetic? Focus on modular arithmetic

Also have a discussion zone, with Google Groups, which is supposed to be for others, though my posts dominate now. Its primary purpose later? Don't send me your math ideas, post! Am a firm believer that public is better, and this person or that? Not going to be that person.

So can look over what I have or if want can post your own at: My Math Group

Math discovered is a human resource. People come and go, but the math? Will fuel further discovery, and be a source of wonder and excitement for human beings, indefinitely.

For as long as there are humans, am sure our species will learn more math.

Discovery is so much fun really.


James Harris

Simply more integer-like numbers

Came across a rather basic proof using simple algebra, relying on quadratics, which shows there are some new numbers which are really cool. Have a post with a proof which will use as a reference in a more general explanation here.

First comfort is will be in complex plane and considering:

P(x) = (g1(x) + 1)(g2(x) + 2)

where P(x) is a primitive quadratic with integer coefficients, g1(0) = g2(0) = 0, but g1(x) does not equal 0 for all x.

That is a generalized quadratic factorization, where IS important functions are normalized. Which just means equals 0 at x = 0. And is deliberately asymmetrical, where I need to remove that to find solutions, but will do something clever, and introduce k, along with new functions.

Introduce k, where k is a nonzero integer, and new functions f1(x), and f2(x), where:

g2(x) = f2(x) + k-2 and g1(x) = f1(x)/k,

multiply both sides by k, and substitute for the g's, which gives me the now symmetrical form:

k*P(x) =  (f1(x) + k)(f2(x) + k)

For here relaxed a requirement that is in the proof, as imagine k = 1 or -1. Then is really just a symbol, not doing anything, but also notice NOW we have symmetry. And I can solve for the f's as roots of the same quadratic, which I can make sure has integer coefficients.

So I can force the f's to be algebraic integers. THAT is way important. With one more function have a handle on things, and made it H(x) for that reason. That is for handle.

And introduce H(x), where I like using the capital letter here for visual reasons, but it is not to signify H(x) must be a polynomial, where:

f1(x) + f2(x) = H(x)

Where yeah, next you just substitute for one of the f's and simplify sensibly. Is VERY easy algebra folks. Hard to imagine how could be much more simple. And get:

f12(x) - H(x)f1(x) - kH(x) - k2 + k*P(x) = 0

So yeah algebra is SO EASY which is good, to remove reasonable doubt! As believe me we are shaking the foundations of number theory. Can you feel it? I can.

For one of my favorite examples have H(x) = 5(7x-1), where have a post where give here. See that yeah, if you have x be integers, then the f's are going to be roots of the same quadratic with integer coefficients which is VERY important.

With k = 1 or -1, we now have a way to find algebraic integer solutions for the original factorization! Way cool. And in fact, with that handle H(x) we can get every possible one, which is where things would end, if we didn't have that clever k to let us do something different.

If we consider k, NOT a unit, so is like k = 2 or even better k = 3, or k = 4, then remember we have:

 g1(x) = f1(x)/k

Which is just algebra.

P(x) = (g1(x) + 1)(g2(x) + 2)

And we now know that one of the g's cannot be an algebraic integer when k is not a unit, for non-rational solutions for the f's where show that in the proof that they must be there. We can force that factor. The algebra doesn't care. So humans thought something wrong for over a hundred years? The algebra is the reality.

Human beings? Can believe anything.

You may wonder, what if we FORCE the g's to be algebraic integers, like take one of our solutions with k = 1 or -1 when we can get all such possible, and THEN do the symmetry thing as well?

The algebra doesn't care, but will NOT let the f's be roots of the same quadratic with integer coefficients then with k not 1 or -1. They will disjoint from each other. What is key is the process where they are roots of the same quadratic with integer coefficients. As far as the math is concerned, you simply have one of the f's multiplied by k, so its removal to get one of the g's is as easy here as anywhere else.

For human beings though may be harder to process what is harder to see direct. The logic is straightforward, but human emotion can get in the way. It can feel off. Which is something physics students face with quantum mechanics worth noting. Your feelings? Don't matter. Logical rules. Do.

Which goes to show you how mathematics can be EASY and very logical, without question where every possibility has an answer. The problem here then with acceptance of the result is entirely social, as it does not get much easier than a quadratic.

These numbers are also integer-like, which means they are like 1+5i which is a gaussian integer, in behaving in ways very much like integers, while being non-rational.

A way more wordy post goes into lots more talk around it all and can be found here. But for example with this approach can prove that 3+sqrt(-26) actually has 7 as a factor for one of its solutions.

Which one? We can't tell. Like 1+sqrt(4) = 3 or -1. Convention can say take the positive, but the math does not care what you, just some human, thinks is ok. The math knows that sqrt(4) is 2 or -2.

So we cannot see direct when can't resolve the square root, and the math doesn't care about that either.

So yeah, these numbers have weird properties too. Which makes sense. Helped them stay hidden until a smart argument looking for them was found. And should be exciting, but number theory operated for over a century with folks not realizing they were there, and social consequences in our time are HUGE.

But the math does not care.

And yeah, did NOT just figure this approach out at random. Was looking for it, thanks to a published proof.

Have questions? Ask away in comments please.


James Harris

Wednesday, October 18, 2017

Another modular inverse example

Will use my method for calculating the modular inverse and put method at end with link to main post on it.

Will find modular inverse of 41 mod 111. Part of the reason picked is 111 = 3(37) as wanted to check with composites presuming primes would be easier cases.

So r = 41 mod 111, using y0 = 1, as usual, and m = 11, so D = 10 and N = 111.

F0 = 41(41 + 2(11)) mod 111 = 41(63) = 30 mod 111

22(30)d = [30(n-1) - 1](41 + 2(11)) mod 111 = [30n - 30 -1](63) = 3n - 66 mod 111

105d = 3n - 66 mod 111

And 3 is a factor can divide through where importantly have to divide the modulus.

35d = n - 22 mod 37, and have a solution with d = 1, and n = 57 mod 37 = 20 mod 37

And turns out that n = 57 mod 111, is what will work next.

r-1 = (57 - 1)(41 + 2(11)) - 2(11)(1) mod 111 = 56(63) - 22 = 65 mod 111

41(65) = 1 mod 111

And learned the hard way when tried n  = 20, and nothing worked, so of course I panicked, until realized equations were working ok. And had to add the modulus once to get to the correct answer modulo 111.

There are things I do just checking things. Like I just tend to pick m2 greater than N for historical reasons. Is necessary? Or even helpful? Does it matter at all? I don't know. And there are possibly easy potential optimizations. An immediate idea is, trying three different values for y0 and picking smallest results for an iterative case. With my small test examples that hasn't been needed, as usually been easy. But have done an iterative example where didn't show all the work. Would think would be more demonstrative with it, but maybe am too emotional about it.

Like I like to say, emotion and math do not go well together. It's just one of the coolest things ever, and I discovered it? Yup. Hard to process. Seems perfect for computer age too. As an algorithm is nothing hard for a computer. For smaller examples can look more tedious than Euclidean method for a human.

Next will put system and link to main modular inverse post.

Here is the system:

 r-1 = (n-1)(r + 2my0) - 2md mod N


Where y0 is chosen as is m, with m not equal to r, and n and d are to be determined. They are found from:

2mdF0 = [F0(n-1) - 1](r + 2my0) mod N

and

F0 = r(r+2my0) mod N

---------------------------------------------------

Copied from my post: Modular Inverse Innovation

It is also derived there.


James Harris

Outlines of the social problem

Mathematics has a certainty which is checkable. And that also means there really is not room for debate when it comes to things proven. And thankfully means that in areas where there is what I like to call a social problem, can likely be explained, logically.

Like in accepted mathematical training students know that you can have a unit paired with a non-unit with integers, for example:

x2 + 4x + 3 = (x+3)(x+1)

However, it is also taught that goes away if NOT rational, and math people teach that with non-rationals you can only have units that remain integer-like, with something like:

x2 + 4x + 1

Where does not factor easily like before. Which is just given as a property of numbers not integers, different from numbers that are integers, when I've proven is a property of the ring of algebraic integers.

Like if I say evens, and then claim that 2 and 6 do not have factors, in evens because 3 is odd. Where evens are not a ring, but is a ready example to highlight how you can easily see a case, where you get something weird. Explanation? Supposedly non-rationals are just different from integers, even when integer-like, as algebraic integers are.

(Oh yeah, so that's NOT a real rule for numbers. With a proper ring, yes, a unit in that ring can always be paired with a non-unit and both be roots of a monic polynomial with integer coefficients. So NO distinction between integers and other integer-like numbers there.)

The proof that there are actually other numbers along with algebraic integers though, just like there are odds along with evens is so easy, I prove it with elementary algebra and also had a prior result related published in a formally peer reviewed mathematical journal, where there is a wild story with the chief editor trying to delete out of the electronic journal after publication and later the journal died.

Also, have piled up mathematical results, including my most recent which is a new way to calculate the modular inverse, which seems built for the computer age as is iterative in a way that works well with computers, so seems perfectly timed for our now. That actually puzzles me a bit. Seems so perfectly timed. Well is here now. And that was months ago.

Takes a LOT to try to ignore something on that level. Humanity only gets so many basic discoveries of that type in ITS entire existence, am sure. There are only so many fundamental results.

Why does the proof of more integer-like numbers impact number theorists in an important way?

Because number theorists are the ones who may operate with ring of algebraic integers, while others, like scientists as also consider beyond mathematicians, tend to use mathematics in fields.

There is no problem in fields. Like notice, if you have 6/2 or 2/6, in a field you can divide through with no problem. As is only with an arbitrary rule where factors matter that you have a problem, like saying only evens.

With ring of algebraic integers, you are saying, only numbers that are roots of some monic polynomial with integer coefficients.

That restriction is arbitrary like saying evens is arbitrary, and the algebra no more recognizes it, than not letting you divide 6 by 2 because you say evens.

So the mathematics is EASY. It is easily proven. And the problem is easily explained, yet over a decade has gone by, and we're past the point of benefit of the doubt for number theorists. Oh yeah and that weirdness with the math journal. Seems like lots of deliberate action.

The best explanation for trying to ignore such a result?

Seems that the simplest explanation is that enough number theorists have research, possibly on which they have based their careers, where it turns out the truth would invalidate that research.

(Do you have a better one? If so, please comment. I'd LOVE to see it.)

Here's the weird thing: if that's true then number theory WOULD crowd. That is, the path into mathematics through number theory would be wider because of the error. And EASE of success would be greater as well. Of course problem is, if try to show that success with actual integers! Unless you tended to avoid them. And greatest success in number theory?

Would probably be for people who MOST used the error. And scariest people?

Applied mathematicians. They would remain ERROR FREE and would need to be sidelined in case they noticed the problem. Or began checking number theorists more closely.

Oh, but of course number theorists DO have one applied area. Math involved in much of computer security globally. Like lots with public key encryption. I hesitate in discussing that area. But it definitely gives pause. Especially if you pay attention to news and routine breaches, which somehow always get explained. But for these people? That may just be what they now accept.

There though WAY out of my area. Just seems so impossible. Requires governments all over the world to just not be paying enough attention, if these people have reached that level of faking it.

Hence the social problem. Now how to solve it?

So far they seem in my opinion to be relying on benefit of the doubt. Looking for people to rationalize around the details.

Which must have worked so far as here we are. And I am still explaining something very obvious, where the mathematics is as absolute as ever. And very simple to explain.


James Harris

Monday, October 16, 2017

Notice to governments

Have come across a problem in mathematics in an important prestigious field, which is number theory. And was able to demonstrate with publication in a formally peer reviewed mathematical journal. The problem is severe enough that the response from the mathematical community was to avoid, including that the chief editor tried to delete the paper out of the electronic publication.

What I did was show that you can present a mathematical argument, correct under ALL mathematical rules established, which gives an incorrect conclusion.

That was back in 2004. When the mathematical community did not act with the appropriate response realized had a bigger problem and endeavored since then to verify my own argument, and was able to so do with secondary means. I found another way to prove the same thing.

Relevance to national governments: mathematicians working for you who are number theorists may not be actually qualified for their positions. This problem can allow them to have invalid research which appears correct by established measures.

You can check. These people are NOT actually effective, of course. But with a problem that has been in the field since the late 1800's they have built a support structure around themselves.

They should not hold security clearances. They should not be relied upon for any mathematical research your government needs.

It is your job as governments to determine when these things are true. Good luck.


James Harris

When apparent success is easy

My own personal understanding of the implications of some of my discoveries is of course important to me, but also have begun to address more my global responsibility. And that global responsibility pushes more care and certainty, which can explain years in working through to be sure of the foundations. Which also worked out great for me, in making more discoveries.

Now though can explain things rather simply, and is my duty to explain simply implications of one of my most important results, which only requires considering a general factorization in the complex plane:

P(x) = (g1(x) + 1)(g2(x) + 2)

where P(x) is a primitive quadratic with integer coefficients, g1(0) = g2(0) = 0, but g1(x) does not equal 0 for all x.

It is trivial to be able to find the g's as algebraic integers, and then step beyond them to other numbers, which are also possible solutions for the g's, which cannot be fractions or like fractions in any way. Which are themselves integer-like, but not previously catalogued.

However, mathematical arguments not recognizing this reality, can appear to prove things NOT true, while looking correct if this reality is ignored.

That in mathematics, as most math students are usually taught, is a ticket to just about anything.

You can with such a problem, potentially appear to prove whatever you want, and my suspicion, since this problem arrived in the late 1800's is that it lead to a shift in the mathematical field towards dominance by people who increasingly exploited it, whether they realized it or not.

Human beings have a knack for finding easier success. Applied mathematicians of course would not be able to exploit it.

If there were not some awareness then my publication of a contradiction back in 2004 would NOT have lead to a math journal imploding, but to a hue and cry, as mathematicians recognized the problem. Instead I've faced what I consider astute use of social things. But for instance, my post giving first a functional definition of mathematical proof, and later updated with a formalized one, was a reaction to mathematicians in the press, diminishing mathematical proof.

There IS a lot of naive I think especially in academic circles with mathematicians who grew up before the web about how well these stories travel. And have been curious about some of the things I suspect or think I notice that are being tried. With a relentless look to the press from certain people with relief as if that is all that matters.

But I do not need the press. Wouldn't mind their help, but don't have to have it.

My duty is to the discipline. Am stepping carefully as I lock down understanding. And my decisions will follow.


James Harris

Sunday, October 15, 2017

Sum two squares and 5

Used my BQD Iterator to highlight the basic result that EVERY power of 10 can be written as sum of two squares, where noticed the squares were even, after the first one where had to use the trivial result that 1 is a square. Well got to wondering--what if divide those powers of 2 off? Here is the result:

42 + 32 = 52

132 + 92 = 2(53)

72 + 242 = 54

792 + 32 = 2(55)

442 + 1172 = 56

3072 + 2492 = 2(57)

Here's the full original list for two squares to powers of 10:

12 + 32 = 10

82 + 62 = 102

262 + 182 = 103

282 + 962 = 104

3162 + 122 = 105

3522 + 9362 = 106

24562 + 19922 = 107

Saturday, October 14, 2017

More power 10 sum two squares

From my research is now known how to easily show how to write every power of 10 as the sum of two squares. Which I think is cool. Thought would post more about it. From last iteration given in my previous post on subject I have:

3162 + 122 = 105

Where have the rule, from my BQD Iterator.

If:  u2 + 9v2 = 10a

Then: (u - 9v)2 + 9(u + v)2 = 10a+1

Noticed that 316, which is an approximation of sqrt(10) times 100, and figure is because other square happened to be small, so am curious, about 107 now.

So will do two more iterations. And can use u = 316 or -316, and v = 4 or -4, so will fiddle with things, which I did. Of course just showing what I decided works ok for my purposes.

(352)2 +  9(312)2 = 106

Which is: 3522 + 9362 = 106

Next will use u = 352, and v = 312:

(-2456)2 + 9(664)2 = 107

Which is: 24562 + 19922 = 107

And didn't go way I thought it might. Had to fiddle with signs and deliberately trying to get a small square again, just ended up getting both sides multiplied by 100. Math does not care what I want. It is interesting to see the math adjust to what you TRY to get it to do. That sense of the math is something. Like there is this intelligence, which I guess, yes it is. It is just perfectly logical.

Guessing though if kept going would find cases where with odd exponent for 10, would again see an approximation of square root of 10 in there. Well, curiosity is satisfied, for now.


James Harris

Thursday, October 12, 2017

When explanation rules

With over 13 years to consider since publication of my paper is not surprising have well-worked explanations now for significance.

What I discovered is if you start by declaring the ring of algebraic integers, have a beginning statement in that ring and take valid algebraic steps from it, you can still reach a conclusion--not true in the ring of algebraic integers!!! Which I now say is like, declaring evens and then dividing 6 by 2 to get 3.

It is a coverage problem, which reveals finally after more than a century since was first believed that roots of monic polynomials with integer coefficients could encapsulate ALL integer-like number that mathematics had remained more subtle than many realized.

My original argument relied on a cubic, but simplified to a quadratic with same result. And even better, found a different path to the same conclusion with the generalized factorization in the complex plane:

P(x) = (g1(x) + 1)(g2(x) + 2)

where P(x) is a primitive quadratic with integer coefficients, g1(0) = g2(0) = 0, but g1(x) does not equal 0 for all x.

Being able to go to the field of complex numbers removed needing to worry about problems with ring of algebraic integers, and definitively reveals the same conclusions as before. A key post is here which steps through the short argument.

So two paths revealing something easily explained in a paragraph, and so much drama around the result, back in 2004. From an academic perspective, with the logical conclusion that NO paper should be declared in the ring of algebraic integers, would guess is a nightmare. Brings into question any mathematical paper declared in that ring. But am NOT an academic. Am a mathematical discoverer, but not a mathematician. So have certain dispassion there.

Is not my only discovery where there has been a sad delay either. Also over a decade ago found a simple approach to counting prime numbers, talk it a lot on the blog, but discuss things fully here, which will reference for this post, giving key points. And my approach tweaks older ideas giving a vastly simpler sieve form:

P(x,n) = [x] - 1 - sum for j=1 to n of {P([x/p_j],j-1) - (j-1)}

where if n is greater than the count of primes up to and including sqrt(x) then n is reset to that count.

But more importantly, the fully mathematized form relies on a difference equation, from which a partial differential equation is readily found. But to make the summation with the difference equation give the correct prime count it has to be limited. You have to deliberately stop it from going to primes above sqrt(x), which of course goes away with the continuous function!

P'y(x,y) = -(P(x/y,y) - P(y, sqrt(y))) P'(y, sqrt(y))

And our connection between the count of prime numbers and a differential form is complete.

Since with the continuous function you're no longer stopping it deliberately, it would keep going, subtracting more, and have a REASON for lagging the count of primes.

And that is what's seen with what we can assume are approximations, with x/ln x, for instance 100/ln 100 equals approximately 21.71, which lags behind the prime count of 25. And 1000/ln 1000 is approximately 144.76 which lags behind the count of primes up to 1000, which is 168. Guessing then it would never catch it.

Does help to be curious though. Am not talking things fresh here, but things have been talking for YEARS. And I wanted to talk both cases because earlier mathematicians had something important--curiousity.

They weren't looking for some way to pursue an academic career.

There was a time when mathematicians were hungry for knowledge.

They were people who really wanted to know things like, how do you cover all integer-like numbers? Gauss had intrigued them with a+bi where 'a' and 'b' are integers. How far could you go? And they DID try. Just turns out mathematics can be wonderfully subtle. Took doing some things different to figure out what I did.

And with prime numbers, Euler, Gauss and yes, Riemann among others were curious!!!

They wondered WHY prime counts could possibly have this seeming connection with continuous functions.

And now we know. Our people in our times have known for years now.

But I guess there is no academic career potential there for others, or things would have gone differently, eh? I disdain such things. So what good is the knowledge then, to the modern mathematician?

Where did that basic curiosity go?

Could talk more things, but feel like have made my point. That I could fill this post with more major discoveries is simply further indictment of a system that lost its way. When mathematicians forgot that the point, is in discovery.

I love discovery. The people who do? Will always be the ones who move things forward, for a planet of humans, who need knowledge.


James Harris

Tuesday, October 10, 2017

Sharing knowledge benefit

Found myself finally citing my own paper proudly and has been a journey of over a decade from that wild story around my one publication, to being able to talk confidently the value of publication as did in a recent post.

There I noted that in sharing truths, we open opportunity to find greater truths.

Of course a person could share for recognition, and readily note that some might think I do, when in the past yeah, maybe more so than now, as I got it. Why push harder for what I already have?

If you think I do not, then you may be looking to certain members of the mathematical community, whom I think may be dedicated to hiding a flaw that may mean they shouldn't actually be considered mathematicians.

Um, why would I expect recognition from them or care if I got it? Oh, power of a position, maybe?

And I did realize that yes, people may be looking at the POSITION even if informed that the person in the position may be behaving a certain way because the truth would remove the position. To me is like a Catch-22 and years ago I shied away from it, and did a mental shrug.

And as for objective evidence, so far last 7 days according to Google Analytics have had visits from 13 countries. US is main one, and not going to list the rest. There is just a constant of countries for me for about a decade. First noticed 2007? Or 2008? And not just for this blog either.

Can watch that jump when I have a new major discovery. Which has happened enough times over around a decade is not remarkable for me. Didn't even bother to closely watch this time with my latest, figuring out my own way to calculate the modular inverse. So no, no problem with recognition.

Oh, does give me the advantage can simply share math things here. Have the reach of a major journal, without the hassles.

Is weird actually. Word I also like to use for how have felt often is, unsettled. Global attention is something you have to learn to process.

People look to the establishment next though, am sure. So I get the recognition all over the globe can watch in web stats, but just not the official from a handful of people I don't think much of anyway.

The web is the new thing that helped me so much, but so much still is being learned, and I can see web stats where others can just see a quiet blog. Which I like.

Sharing for opportunity to learn more though is a concept that has inspired me. Yes, have been sharing that way for some time but hadn't written it out, just that way.

We know what we know as humans because of knowledge shared. And now I take it less for granted than I did before.


James Harris

Monday, October 09, 2017

Five months since modular inverse discovery

No doubt my discovery of a third primary way to find the modular inverse, where other two have Euclid's name on one and Euler's on the other, impacted my thinking. And was a surprise for me as had decided to end my basic research phase, well satisfied with how things had gone.

Here's the post, though have put it up quite a bit since then: Modular inverse innovation

That THRILL of discovery is hard to describe. And I realized that yeah, society does believe that researchers in a field know and have it with discoveries in their area. And is very motivating, I assure you.

But what if you don't have it?

Well, my guess is, you will find it hard to understand, what you do not know.

For some who may think they are mathematical discoverers, like me, math is about recognition from peers, and is a social experience only.

For me though will not contain my enthusiasm as yeah, still excited over it. At times just ponder, and wonder.

Shows value of a great math education--where guess mine was better than I realized till recently--where smart teachers let young people discover things along the way, versus just dictating knowledge to them.

The thrill of discovery is amazing. Those who have it? Just wish all to feel it who can.

I have MY own way to calculate the modular inverse.

When you feel that thrill, yeah is motivating. Can be addictive too.

That perspective is helping me more as shift from a more competitive view, and realize DO need to help others understand that potential.

What can YOU discover? The math is there to find. Believe that.


James Harris

Sunday, October 08, 2017

Publication does matter

Much easier to talk value of publication in a peer reviewed mathematical journal now, versus over a decade ago, when I was published in what I like to call a wacky story. The chief editor tried to literally pull the paper after publication by simply deleting out from the electronic file, and claimed it was withdrawn, when I never withdrew it.

The paper is "Advanced Polynomial Factorization", published in the Southwest Journal of Pure and Applied Mathematics, Issue 2, December 2003, pp. 6–8.
Submitted: July 25, 2003. Published: December 31 2003

Copied from my post, where also is a link to paper: Publishing a contradiction

And EMIS simply put up a copy of my original, and is worth talking why publication matters which to me is NOT for recognition.

To me the value of publication is in sharing truths which open opportunity to finding greater truths.

Where that opportunity is like an open door, and for me was a joy in going through it, where have simplified that original argument, gained a greater understanding of mathematics in the process, and found a path to even greater knowledge.

We share best for each other, and human progress has occurred because of vast quantities of valuable shared information. That some person merely kept hidden away, which disintegrated in time never to be seen? Is not part of our shared human story. But that is a choice one can make.

Maybe the reaction I got was telling, back then. Some howled in fury, maybe helping push that chief editor. Others decried the supposed horrible impact on the math discipline, oh horror, as if. And many dismissed the math journal process as a broken joke.

So yeah, much easier to talk calmly and with objectivity, with experience, after over a decade. Math and emotion do not go well together, I like to say.

When two anonymous reviewers--as the now defunct journal used two--did the right thing, looking at correctness over social implications. And EMIS has done the right thing, preserving information rather than putting on a social filter.

And society is better for people who understand the value of process and opportunity given, rather than simply looking to take opportunity away.

The math does not change, regardless.

Share best for opportunity to learn more, in my opinion. And the journal process will do just fine, as long as keeps opening doors revealed from shared truths, to further greater knowledge.


James Harris

Saturday, October 07, 2017

Some consequences

Some of my research forces troubling conclusions with regard to mathematical community, especially number theorists, where is useful to step through relevant math again.

For instance now have noted ability to determine much from a generalized factorization.

In the complex plane:

P(x) = (g1(x) + 1)(g2(x) + 2)

where P(x) is a primitive quadratic with integer coefficients, g1(0) = g2(0) = 0, but g1(x) does not equal 0 for all x.

And I went in the direction of multiplying both sides by some integer k, introducing new functions to gain symmetry, and showed could find those functions as algebraic integer roots:

k*P(x) =  (f1(x) + k)(f2(x) + k)

Where show all in this post, but primary point is that in so doing, could prove a coverage problem with the ring of algebraic integers. And it's not even complicated. Is like if you SAY you only want to use evens, and then have a mathematical argument where you claim that 2 is coprime to 6, because you've excluded 3 which is odd. But of course that's specious, as SAYING to use only even numbers does not change the actual reality.

Across number theory this problem may emerge if there is a claim of a unit factor in ring of algebraic integers. Where algebraically with a more complete ring, without this problem, that is clearly not the case. And I had a more complex argument showing the same thing with cubics and an entirely different approach which was published in a math journal. Talked a bit about here.

The problem of course is you may have some number theorist who may have research which is invalidated by this result, when he thought he had valuable contributions to human knowledge. And that could be key to his social status, I hypothesize. And maybe there were enough people like that where they decided to run from the math, and survive in their positions.

Given the amount of time that has passed since I first pointed out the problem, with a published paper, where the journal chief editor tried to pull after publication and journal shut down, maybe he was one of them. And if were me, like to think I'd want the truth, but is not me. I'm the discoverer.

Looking across human systems it actually would have been more remarkable if they'd accepted the truth. Which is kind of sad that what they did instead is not surprising, and is continuing.

Of course you may have students being taught by people without real mathematical accomplishment, so how can they guide to any? And learning erroneous approaches which means can have invalid mathematical arguments on which they build their OWN careers, when house of cards will collapse eventually.

Or, can simply say, best guess is an unknown number of these number theorists could be simply frauds. And it's not like they can teach what they do not know, you know? Discovery IS hard. The coverage problem gives an opening for people who might never experience that adulation, acceptance, and social status, any other way, because real mathematics can be just WAY hard.

Also it's just sad. Have pointed to number authority recently as math can be very enjoyable, and satisfying with RESULTS where in number theory, yeah you can do cool things with real ones.

Like check this out:

(922 + 962 + 1442)(70482 + 6882 + 1032+ 1720+ 24082) = p2 + 1231q2 

Where p = 1507976, and q = 4920 are solutions. And I actually relied on this result for a post about number examples I like to stare at, where post with it is, here.

Why does society allow deliberate and continuing error from powerful, influential people at high levels in very important positions?

I think that is a GREAT question. Consequences are so HUGE. For me though? Not so much. I just kept discovering, after all. For the discoverer? Is just more information. Knowledge obtained.

My place in history, after all, is guaranteed. Rest of you? Are competing for some kind of a place.

Where for lots of you? Just is not going to happen.

Just not competing very well. For many of you, this result alone guarantees that nothing you do in mathematics, assuming having that mathematical audience, will be taken seriously without wondering where were you, in this tragedy of the mathematical world?

While I've been talking it, for years.

Oh yeah, took some time for me to get perspective. And weird thing? These kind of stories spread FAST, so lots of people must know by now. But then they do nothing much. Which is weird to me I guess. But watching with other stories in other areas, apparently are waiting for an appropriate authority to handle?

Which to me? Is kind of interesting. So yeah, some of you math people? They DO know. The Public.

And when they look at you? What do you think they're thinking?


James Harris