Well may as well baseline current somewhat. While also pondering what may be doing going forward.
One of the coolest things early was figuring out a simple way to count prime numbers, which I talk endlessly; relies yes on previously known ideas, but with a slight but powerful tweak. And not only had my own way to count primes, eventually realized lead to a partial differential equation. Which did have me at one point actually reading Riemann's notes, in original German as studied some. Was an odd feeling there but that was over a decade ago.
My find of a coverage problem showed the efficacy of my tautological spaces which also helped with a new way to reduce binary quadratic Diophantine equations. Have expanded on tautological spaces so are general so can tackle other things. And have my ring of objects. Actually may as well admit am usually doing things in the ring of objects, though not declared.
There is my prime residue axiom. And also have some additional tools for probing numbers where the BQD Iterator is one of my most widely used. Amazes me how versatile it is.
Where weirdly played with VASTLY more after giving a cool name. Had had for years. What's up with that? Naming things. Who knew?
There are plenty of things with which can occupy my time when I feel like it. Or simply review, thinking to myself--I found that.
Oh and of course still trying to emotionally grasp having my own modular inverse method. That just came like a bolt from the blue last year.
Am lucky. The research has simply been rewarding. Am a solitary researcher where don't even pretend otherwise but maybe in the past didn't so boldly declare. Now find that comforting to state clearly. And don't have to do things would never have wanted to do, like be an academic.
Is so perfect a situation in certain ways from my perspective. Yet at times that makes me wonder.
The math isn't going anywhere. Once found and shared, is simply available for our species.
Clearly have to a large extent relished controversy around my research, where guess that hasn't been secret. And been grateful to quit even bothering with trying to talk to mathematicians, despite occasional things here or there where feel a duty. If up to me? Would be done with them for good. As well as emphasizing am not one and will not become one. The math is the draw. Web enables people to get it when they need it.
Other things? Not needed. And now would simply be tedious, as going through motions needed in a world that has gone away.
Even now I work more to handle the constant global attention. More aware I know than others.
Which actually is true. Means have to totally let go of blaming math people for other problems which are just about me, like with money. The web does free you up from needing all that academic machinery including journals if you're NOT into it and NOT an academic, so don't need the silos which were needed in the past.
Yet it also pushes you to rapidly figure out how to then do ok, without the financial support. But at least the lack of approval from mathematicians have deemed irrelevant. Public does NOT notice or care. Just can't really talk math to them anyway. And why bother? Let interest drive people here.
Well enough for a basic baseline. Finishing up things has me better defining my research as abstract reductionism realized. And much of what I do is dominated by modular. And having introduced procedures that have given the world's first true modular algebra, there is so much can do both there and beyond.
Is SO cool.
Still do have responsibilities feel like to mathematical industry is how I like to phrase. And have stated will revisit in 2028, God willing. And if necessary.
Now though with so much settled can get back to just enjoying a continuing conversation with the math. More contemplating what I already have. NO plans on active research.
Have had enough fun there.
James Harris
Monday, January 29, 2018
Saturday, January 27, 2018
Social problem evaluation complete for a decade
As much as I think information travels is not necessarily the case that academic mathematicians are aware of my research. Even with sending by email am not sure was read.
Which is ok. If academics do not know that would of course put academics off the hook! And at this point in time is not readily something can evaluate.
My past evaluations are pending waiting for more information. So they will remain in place.
So will let more time pass. And God willing, will return to this concern in a decade if necessary.
Which lets me return blog to usual type postings, so will let this post end current investigation into the social problem.
So next posting on that subject, noting to self, should not be before January 2028.
Blog will now return to math subjects as interests me, without worrying on such matters.
James Harris
Which is ok. If academics do not know that would of course put academics off the hook! And at this point in time is not readily something can evaluate.
My past evaluations are pending waiting for more information. So they will remain in place.
So will let more time pass. And God willing, will return to this concern in a decade if necessary.
Which lets me return blog to usual type postings, so will let this post end current investigation into the social problem.
So next posting on that subject, noting to self, should not be before January 2028.
Blog will now return to math subjects as interests me, without worrying on such matters.
James Harris
Friday, January 26, 2018
Social implications from modular inverse discovery reception
One of the more important recent events for me and also for mathematical research was my find of a new primary way to calculate the modular inverse, which was posted here May 5th 2017. With such a major research find was able to settle any number of questions in the social domain as well which helped me evaluate modern academia.
What is easily established is: is a major discovery without concern of validity, is a surprising discovery as a primary in this area is a surprise, and there is no justification possible for mathematicians of any type, either applied or pure to ignore.
Also the result in and of itself has pure and applied mathematical characteristics, so in essence, is a perfect litmus test across multiple categories.
And the mathematical academic world failed that test.
Will note my diligence as a researcher and discoverer did mean again at least sending to some academic mathematicians and also to the NSA, twice. No point in naming the few mathematicians though will admit at least some were at a University of California school.
There was no response from academics to emails. NSA responded form letter to first effort submitted through their website. Did not respond to second effort.
Regardless as is usual with my mathematical discovery, web search has been more revealing, and there are searches which have trended is the appropriate phrase telling myself, higher. Will not post the searches am checking in this post as that can impact their behavior which am still studying. Though have noted elsewhere as also experimented by posting on Reddit, which have discussed on this blog. And primarily attracted trolls.
This failure of the mathematical community is fascinating to me though. And DID explore more what I call the social problem as considered it. However, is kind of easy to understand from an evolutionary and anthropological perspective. Am not sure on that phrasing. Is easy if you consider humans as very much an ape species, which we are. Is too bad the truth revealed by science also has political tones. And our evolution beyond other ape species is shallow in many ways. Where intellectual behavior is involved with highest brain function, but am looking at purely naive territorial behavior.
Like it ignores my ability to just talk out situation on web. Also ignores likely impact across academia which will likely be vastly larger than just for mathematicians. That ignores potential impact for mathematicians for generations in academic world as well.
Academic process arrived to a large extent with Sir Isaac Newton in its more modern form. Though also much remains from prior am sure. Am not surprised a revisiting from top to bottom may be in order. Leverage from this result could be significant in that process.
Yup, may be an opportunity to remake the modern university. Cool! Time will tell.
To me as a mathematical discoverer though is an interesting challenge to mathematical truth.
Which definitely indicates shows behaviors rooted in lower brain function because higher thought of course realizes how this story will end. The mathematics will win.
In the meantime will admit situation allowed me to relax considerably. Cleaned up a few things including settling my 14 years or so leisurely consideration of coverage problem with algebraic integers, which also lets me champion my fascinating quadratic factorization number theoretic probing tool.
While also continuing other activities which actually consume most of my time now. Is not like there is much pressure on me from a research perspective. Now I have more practical concerns often. So I just get to enjoy discovered things. Is cool.
James Harris
What is easily established is: is a major discovery without concern of validity, is a surprising discovery as a primary in this area is a surprise, and there is no justification possible for mathematicians of any type, either applied or pure to ignore.
Also the result in and of itself has pure and applied mathematical characteristics, so in essence, is a perfect litmus test across multiple categories.
And the mathematical academic world failed that test.
Will note my diligence as a researcher and discoverer did mean again at least sending to some academic mathematicians and also to the NSA, twice. No point in naming the few mathematicians though will admit at least some were at a University of California school.
There was no response from academics to emails. NSA responded form letter to first effort submitted through their website. Did not respond to second effort.
Regardless as is usual with my mathematical discovery, web search has been more revealing, and there are searches which have trended is the appropriate phrase telling myself, higher. Will not post the searches am checking in this post as that can impact their behavior which am still studying. Though have noted elsewhere as also experimented by posting on Reddit, which have discussed on this blog. And primarily attracted trolls.
This failure of the mathematical community is fascinating to me though. And DID explore more what I call the social problem as considered it. However, is kind of easy to understand from an evolutionary and anthropological perspective. Am not sure on that phrasing. Is easy if you consider humans as very much an ape species, which we are. Is too bad the truth revealed by science also has political tones. And our evolution beyond other ape species is shallow in many ways. Where intellectual behavior is involved with highest brain function, but am looking at purely naive territorial behavior.
Like it ignores my ability to just talk out situation on web. Also ignores likely impact across academia which will likely be vastly larger than just for mathematicians. That ignores potential impact for mathematicians for generations in academic world as well.
Academic process arrived to a large extent with Sir Isaac Newton in its more modern form. Though also much remains from prior am sure. Am not surprised a revisiting from top to bottom may be in order. Leverage from this result could be significant in that process.
Yup, may be an opportunity to remake the modern university. Cool! Time will tell.
To me as a mathematical discoverer though is an interesting challenge to mathematical truth.
Which definitely indicates shows behaviors rooted in lower brain function because higher thought of course realizes how this story will end. The mathematics will win.
In the meantime will admit situation allowed me to relax considerably. Cleaned up a few things including settling my 14 years or so leisurely consideration of coverage problem with algebraic integers, which also lets me champion my fascinating quadratic factorization number theoretic probing tool.
While also continuing other activities which actually consume most of my time now. Is not like there is much pressure on me from a research perspective. Now I have more practical concerns often. So I just get to enjoy discovered things. Is cool.
James Harris
Thursday, January 25, 2018
Probing number properties with quadratic factorization
Feel extremely lucky that a simple quadratic factorization was available for checking a number of things, and was around 2010 started studying. Will talk how that basic factorization allows probing deeply into numbers.
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 is VERY important that the g's are normalized. While studying with algebraic integer solutions may have been done, came up with a generalized way to do that--and beyond.
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)
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
Which gives me total power to force a monic quadratic with integer coefficients by how I choose k and H(x). 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, with k = 1 or -1, you just get algebraic integers if you have H(x) be a polynomial with integer coefficients.
However, you now have that if k is some other nonzero integers besides 1 or -1, then one of the g's can be blocked from being an algebraic integer by: g1(x) = f1(x)/k
That blocking of course occurs when non-rational.
But the product then is still allowed to be an algebraic integer: f1(x) = kg1(x)
The entanglement with non-rationals is FASCINATING. And yet, k is proven to be a factor.
Here should note is where you can introduce contradiction! If instead of declaring in the complex plane, could get contradiction or not if declared in ring of algebraic integers just by choice of k.
The main point there is simple: one of the g's provably cannot be an algebraic integer if nonzero integer k is not 1 or -1, and the g's are not rational, when H(x) is a polynomial with integer coefficients. While one of the g's IS being forced to be an algebraic integer.
Where simplicity in the proof comes from the f's being roots of the same quadratic. H(x) is key here. Depending on how is chosen you are then pushing back to pick the g's.
Which is how the analyst is choosing from infinite possibility.
Should admit was EXTREMELY surprised that the algebra DOES recognize as distinct roots of monic polynomials with integer coefficients i.e. algebraic integers.
And did not fully appreciate against my biases until this post November 2015, when finally considered k = 1 or -1. Where had been considering algebraic integers as just kind of a mistake since my own research from 2003, so had around 12 years with that idea in mind. But the math actually has them as distinct and is curious in what ways versus other members of my ring of objects.
Where at least know that there are other members NOT algebraic integers where multiplying times some non-unit, nonzero integer gives a product that is.
Oh so yeah, bringing up my ring of objects which includes all integer-like numbers as well as integers themselves. And my ring of objects excludes any numbers not integer or integer-like, which was the approach I used to include them all. Switching to abstraction of exclusion to include versus attempting to get all with rules of inclusion.
So much from a basic factorization! And also feel lucky and amazed with the power of abstraction shown with this approach! Finding is one thing, but is great for that mathematical logic to be there.
These are mathematical pieces giving a handle on some aspect of infinity.
That is wild. To me H(x) is fascinating with that ability to handle. It is meant to cover an infinite number of possible expressions where as a human researcher am focused very narrowly. Getting a handle on infinity in this way intrigues me as a powerful technique.
Having a quadratic factorization available then leads to an easy way to thoroughly check on number theoretic properties.
Reference post: Simple Generalized Quadratic Factorization
James Harris
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 is VERY important that the g's are normalized. While studying with algebraic integer solutions may have been done, came up with a generalized way to do that--and beyond.
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)
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
Which gives me total power to force a monic quadratic with integer coefficients by how I choose k and H(x). 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, with k = 1 or -1, you just get algebraic integers if you have H(x) be a polynomial with integer coefficients.
However, you now have that if k is some other nonzero integers besides 1 or -1, then one of the g's can be blocked from being an algebraic integer by: g1(x) = f1(x)/k
That blocking of course occurs when non-rational.
But the product then is still allowed to be an algebraic integer: f1(x) = kg1(x)
The entanglement with non-rationals is FASCINATING. And yet, k is proven to be a factor.
Here should note is where you can introduce contradiction! If instead of declaring in the complex plane, could get contradiction or not if declared in ring of algebraic integers just by choice of k.
The main point there is simple: one of the g's provably cannot be an algebraic integer if nonzero integer k is not 1 or -1, and the g's are not rational, when H(x) is a polynomial with integer coefficients. While one of the g's IS being forced to be an algebraic integer.
Where simplicity in the proof comes from the f's being roots of the same quadratic. H(x) is key here. Depending on how is chosen you are then pushing back to pick the g's.
Which is how the analyst is choosing from infinite possibility.
Should admit was EXTREMELY surprised that the algebra DOES recognize as distinct roots of monic polynomials with integer coefficients i.e. algebraic integers.
And did not fully appreciate against my biases until this post November 2015, when finally considered k = 1 or -1. Where had been considering algebraic integers as just kind of a mistake since my own research from 2003, so had around 12 years with that idea in mind. But the math actually has them as distinct and is curious in what ways versus other members of my ring of objects.
Where at least know that there are other members NOT algebraic integers where multiplying times some non-unit, nonzero integer gives a product that is.
Oh so yeah, bringing up my ring of objects which includes all integer-like numbers as well as integers themselves. And my ring of objects excludes any numbers not integer or integer-like, which was the approach I used to include them all. Switching to abstraction of exclusion to include versus attempting to get all with rules of inclusion.
So much from a basic factorization! And also feel lucky and amazed with the power of abstraction shown with this approach! Finding is one thing, but is great for that mathematical logic to be there.
These are mathematical pieces giving a handle on some aspect of infinity.
That is wild. To me H(x) is fascinating with that ability to handle. It is meant to cover an infinite number of possible expressions where as a human researcher am focused very narrowly. Getting a handle on infinity in this way intrigues me as a powerful technique.
Having a quadratic factorization available then leads to an easy way to thoroughly check on number theoretic properties.
Reference post: Simple Generalized Quadratic Factorization
James Harris
Wednesday, January 24, 2018
Pragmatic research perspective works
Coming into doing my own looking for mathematics for fun was a great thing for me. So yeah there was never any reason to think would be collaborating with others as for all I knew would never find anything. And am NOT a mathematician, but am someone interested in numbers who has had a lot of of modern problem solving training.
When I DID have a profound find and dutifully went to mathematicians was kind of thinking certain things would happen which did not.
And I got to do solitary research exploring my own major research finds and gain a different perspective not just on the mathematical community, but also academia.
With a need to know, the pursuit of truth was fun for me, and still is. So no problem there. Actually am type to do solitary research, so it just went well.
However DID learn some things. And in my opinion many modern academics are desperate for that one thing or other which will give them a burst of attention, usually with others in their field, but SOMETIMES you can get news headlines, on which to build a career.
And academics are terrified of looking bad, or upsetting the majority so work very hard to be sure to never get on the wrong side of their colleagues, especially with research interests, which I think is sad and maybe problematic. The truth does not care. What isn't discovered as a result?
Which to me is the celebrity mentality. Could replace academic with singer or actor, and is the same dynamic.
Looking for that Big Break. Wanting to exploit for as much as you can to get status. Then milk that status for all its worth where academics DO get an advantage there in terms of longevity with it. Though for celebrities can kind of get equivalent, like with a Grammy Award or an Academy Award, which can tout for a lifetime.
That to me explains what I call the social problem--like full mathematicians with advanced degrees, behave as if social opinion is all that matters.
Yeah for THEIR careers.
So I learned functionally reality of what it is like to be a mathematician. And no, do not want to be a mathematician.
Am thankful escaped ever thinking I might want to be.
Almost became a physicist, and yeah, glad simply escaped modern academia.
The ideal of academia is giving people freedom to pursue best ideas to further human knowledge. And much of that does get done am sure. My support of academia remains though more is with love of the ideal will admit.
In my opinion, modern academia presented to the public is more often defined by people looking for a career than people searching for truth. And it shows.
Public is ever more skeptical, as am I, as am part of the public too. Am NOT an academic and have no interest in becoming one. And am far more testing of the reality now.
The wannabe celebrity types? Are so easy for me to spot now.
James Harris
When I DID have a profound find and dutifully went to mathematicians was kind of thinking certain things would happen which did not.
And I got to do solitary research exploring my own major research finds and gain a different perspective not just on the mathematical community, but also academia.
With a need to know, the pursuit of truth was fun for me, and still is. So no problem there. Actually am type to do solitary research, so it just went well.
However DID learn some things. And in my opinion many modern academics are desperate for that one thing or other which will give them a burst of attention, usually with others in their field, but SOMETIMES you can get news headlines, on which to build a career.
And academics are terrified of looking bad, or upsetting the majority so work very hard to be sure to never get on the wrong side of their colleagues, especially with research interests, which I think is sad and maybe problematic. The truth does not care. What isn't discovered as a result?
Which to me is the celebrity mentality. Could replace academic with singer or actor, and is the same dynamic.
Looking for that Big Break. Wanting to exploit for as much as you can to get status. Then milk that status for all its worth where academics DO get an advantage there in terms of longevity with it. Though for celebrities can kind of get equivalent, like with a Grammy Award or an Academy Award, which can tout for a lifetime.
That to me explains what I call the social problem--like full mathematicians with advanced degrees, behave as if social opinion is all that matters.
Yeah for THEIR careers.
So I learned functionally reality of what it is like to be a mathematician. And no, do not want to be a mathematician.
Am thankful escaped ever thinking I might want to be.
Almost became a physicist, and yeah, glad simply escaped modern academia.
The ideal of academia is giving people freedom to pursue best ideas to further human knowledge. And much of that does get done am sure. My support of academia remains though more is with love of the ideal will admit.
In my opinion, modern academia presented to the public is more often defined by people looking for a career than people searching for truth. And it shows.
Public is ever more skeptical, as am I, as am part of the public too. Am NOT an academic and have no interest in becoming one. And am far more testing of the reality now.
The wannabe celebrity types? Are so easy for me to spot now.
James Harris
Monday, January 22, 2018
Basic coverage problem research overview
My most challenging result thought would have help from mathematicians, which is key, as am NOT a mathematician as I routinely emphasize. And publication seemed like a good start, back about 14 years ago when a key paper highlighting the ability to appear to prove something not true with valid methods, disprovable by established number theory finally went live. The problem I'd found there was the declaration--in the ring of algebraic integers.
And talked much in detail in 2017 with post: Publication does matter
Paper is simply correct declared in the complex field without even apparent error. And importantly is also correct declared in my ring of objects.
Unfortunately the rest of the story made things more challenging in some ways, while being a boon for me from a singular research perspective. As have discussed in detail and chase the link to get more links to full story, the chief editor tried to withdraw my result, by simply deleting out of electronic file of an electronic journal. EMIS maintains the original, and I moved on.
This post will cover my primary basic research in that area primarily, as without help from the mathematical community had the vastly more difficult task of personal verification.
One area where focused was: how could I know for sure on my own? So I found a functional definition of mathematical proof and promptly put to use.
Importantly simplified also from cubics to quadratics. And first public post on this blog I note for here where that has been shown to have happened is with my wrapper theorem post in August 2007.
Wasn't until September 2008 that I further tested my use of what I call tautological spaces, which are complex identities, with my Quadratic Diophantine Theorem post, where used conditional:
c1x2 + c2xy + c3y2 = c4z2 + c5zx + c6zy
where the c's are constants, and importantly, realized could set z=1, and got my method for reducing binary quadratic Diophantine equations. So yeah, there is a research path with three variables which have not pursued.
It was by 2010 that I had the idea of focusing on a simple quadratic factorization:
P(x) = (g1(x) + 1)(g2(x) + 2)
Is interesting then had about six years to figure out how should be constructed, where yeah that 1 and 2 are benefits of research, though also kind of obvious to pick?
But looks like don't have a public post discussing until February 2015, with post talking blocking with algebraic integers where had to correct recently. However I also did my most exhaustive stepping through of original approach in the paper now using quadratics in September 2011, with post: Under the hood
And am skipping other research areas related as trying to be more focused for this post. And is interesting that gap I think from having a second path to proving the coverage problem, to what found public now on this blog.
But even then didn't really push it much until last year.
So have roughly a span of research in this area of about 14 years from point of publication which is not extraordinary I don't think. And was allowed to figure things out at a rather leisurely pace, while was also doing other research.
But that was just my own validation. I wanted to be thorough.
From perspective of a solitary researcher really cannot think of it flowing any better. Though it is intriguing was given the opportunity, which is outside scope of this post.
James Harris
And talked much in detail in 2017 with post: Publication does matter
Paper is simply correct declared in the complex field without even apparent error. And importantly is also correct declared in my ring of objects.
Unfortunately the rest of the story made things more challenging in some ways, while being a boon for me from a singular research perspective. As have discussed in detail and chase the link to get more links to full story, the chief editor tried to withdraw my result, by simply deleting out of electronic file of an electronic journal. EMIS maintains the original, and I moved on.
This post will cover my primary basic research in that area primarily, as without help from the mathematical community had the vastly more difficult task of personal verification.
One area where focused was: how could I know for sure on my own? So I found a functional definition of mathematical proof and promptly put to use.
Importantly simplified also from cubics to quadratics. And first public post on this blog I note for here where that has been shown to have happened is with my wrapper theorem post in August 2007.
Wasn't until September 2008 that I further tested my use of what I call tautological spaces, which are complex identities, with my Quadratic Diophantine Theorem post, where used conditional:
c1x2 + c2xy + c3y2 = c4z2 + c5zx + c6zy
where the c's are constants, and importantly, realized could set z=1, and got my method for reducing binary quadratic Diophantine equations. So yeah, there is a research path with three variables which have not pursued.
It was by 2010 that I had the idea of focusing on a simple quadratic factorization:
P(x) = (g1(x) + 1)(g2(x) + 2)
Is interesting then had about six years to figure out how should be constructed, where yeah that 1 and 2 are benefits of research, though also kind of obvious to pick?
But looks like don't have a public post discussing until February 2015, with post talking blocking with algebraic integers where had to correct recently. However I also did my most exhaustive stepping through of original approach in the paper now using quadratics in September 2011, with post: Under the hood
And am skipping other research areas related as trying to be more focused for this post. And is interesting that gap I think from having a second path to proving the coverage problem, to what found public now on this blog.
But even then didn't really push it much until last year.
So have roughly a span of research in this area of about 14 years from point of publication which is not extraordinary I don't think. And was allowed to figure things out at a rather leisurely pace, while was also doing other research.
But that was just my own validation. I wanted to be thorough.
From perspective of a solitary researcher really cannot think of it flowing any better. Though it is intriguing was given the opportunity, which is outside scope of this post.
James Harris
Friday, January 19, 2018
Some overview early 2018
Last year was great for me as managed to settle some things, and now am focused on posts that assume people who read can check math versus me focused at all on any other assumption.
My mathematical approaches are focused simply, with most requiring only knowledge of algebra and modular algebra, which aids in checking. And with me is a true modular algebra versus what I notice primarily elsewhere is modular arithmetic.
And have had a powerful technique since December 1999, which involves subtracting conditional expressions from a tautology which is a complex mathematical identity, like:
x+y+vz = 0(mod x+y+vz)
Of course then there is no doubt about correctness as long as get each mathematical step correct. And made an absolute proof demo post where prove that:
If: x2 + y2 = z2 then: (v2 - 1)z2 - 2xy = 0(mod x+y+vz), where v can be any value.
With much discovery using this technique and following from it, have gained confidence with it, and there is that troubling area of why others have not cheered to my knowledge.
But had a clue from decades ago when a modular approach to packing of spheres was summarily rejected when I sent to the Proceedings of the AMS back in 1996. Long before I was very much aware of a coverage problem with ring of algebraic integers in 2003. Best explanation is that modern number theorists long ago started resisting valid mathematical discovery! Which explained much last year.
Now that is so great a resistance they are not acknowledging a new primary modular inverse method I found last year, and my analysis indicates their focus is on: control the mathematics by controlling public opinion of the discoverer. Where originally I dismissed that approach as meaningless.
My focus is on the math. For a long time I scoffed at the notion that my credibility mattered as it does not as to correctness. But DOES matter as to people checking! As yes, can be a problem if assertions are simply dismissed so mathematical proof is not considered.
The idea of a democratic aspect to mathematical industry I find distasteful. But then again, mathematical industry is dynamic not just on what works, but on people knowing what works.
That number theorists, who I think are the pure mathematicians who are the primary culprits, can fight mathematical discovery is not as surprising to me now, as human behavior in this area is well-known. In this case my best guess is that government funding for research that is invalidated by valuable and valid discovery is a major part of it. So the monetary motivation provides easy explanation for much.
Which seemed obvious to me over a decade ago, but is wacky enough was worth carefully checking that assessment. Is remarkable though, how a system designed to facilitate pursuit of human knowledge was turned upside down.
It is of interest that going to applied mathematicians does not break that as they are cowed and outnumbered. Beaten down I think deliberately by people who knew somehow before their research was usually worthless. And now know mathematically why, from me! That story is SO wild. I do love it, do admit.
Is now of interest how things play out as time marches on.
Mathematics contained? Intriguing to even think possible, eh? I think so.
James Harris
My mathematical approaches are focused simply, with most requiring only knowledge of algebra and modular algebra, which aids in checking. And with me is a true modular algebra versus what I notice primarily elsewhere is modular arithmetic.
And have had a powerful technique since December 1999, which involves subtracting conditional expressions from a tautology which is a complex mathematical identity, like:
x+y+vz = 0(mod x+y+vz)
Of course then there is no doubt about correctness as long as get each mathematical step correct. And made an absolute proof demo post where prove that:
If: x2 + y2 = z2 then: (v2 - 1)z2 - 2xy = 0(mod x+y+vz), where v can be any value.
With much discovery using this technique and following from it, have gained confidence with it, and there is that troubling area of why others have not cheered to my knowledge.
But had a clue from decades ago when a modular approach to packing of spheres was summarily rejected when I sent to the Proceedings of the AMS back in 1996. Long before I was very much aware of a coverage problem with ring of algebraic integers in 2003. Best explanation is that modern number theorists long ago started resisting valid mathematical discovery! Which explained much last year.
Now that is so great a resistance they are not acknowledging a new primary modular inverse method I found last year, and my analysis indicates their focus is on: control the mathematics by controlling public opinion of the discoverer. Where originally I dismissed that approach as meaningless.
My focus is on the math. For a long time I scoffed at the notion that my credibility mattered as it does not as to correctness. But DOES matter as to people checking! As yes, can be a problem if assertions are simply dismissed so mathematical proof is not considered.
The idea of a democratic aspect to mathematical industry I find distasteful. But then again, mathematical industry is dynamic not just on what works, but on people knowing what works.
That number theorists, who I think are the pure mathematicians who are the primary culprits, can fight mathematical discovery is not as surprising to me now, as human behavior in this area is well-known. In this case my best guess is that government funding for research that is invalidated by valuable and valid discovery is a major part of it. So the monetary motivation provides easy explanation for much.
Which seemed obvious to me over a decade ago, but is wacky enough was worth carefully checking that assessment. Is remarkable though, how a system designed to facilitate pursuit of human knowledge was turned upside down.
It is of interest that going to applied mathematicians does not break that as they are cowed and outnumbered. Beaten down I think deliberately by people who knew somehow before their research was usually worthless. And now know mathematically why, from me! That story is SO wild. I do love it, do admit.
Is now of interest how things play out as time marches on.
Mathematics contained? Intriguing to even think possible, eh? I think so.
James Harris
Thursday, January 18, 2018
Finding knowledge rules
We get born and get told things. And to me knowledge is useful information and folks in the modern era get told PLENTY. However, humans found that knowledge, right? And the process can have problems, where we have humans at the front lines called scientists as well as others who keep figuring things out.
To me is an awesome process and definitely respect while also find it fascinating when people are told facts and believe you are trying to convince them of something.
In mathematics there is mathematical proof. And proof is definitive.
If knowledge is available, yet you believe things not true, where that can be determined by fact, then I suspect you have problems with determination of fact, or have not been informed.
With those able to access the web, who have training will admit my belief is: ability to determine fact when available represents a valuable skill.
And I also believe that some people lack ability to determine fact and simply rely on what they are told. Which is NOT necessarily a terrible thing! Most people must simply rely upon what they are told for MUCH. Like with medicine, am going to listen to that medical doctor. Probability can challenge with authority and better my own health versus endanger? Very small. Or nonexistent.
Our human reality is vastly complex with too much knowledge for any one human to have much of a grasp of it all. However, it is still possible to determine fact, if you have the skills. While some apparently require the majority to believe fact, when is determined or they may instead believe false information which to me is simply, not knowledge.
This blog audience focus from now on is to be directed towards those who can determine mathematical fact.
The ability to find knowledge should be cheered and supported. And in my analysis people who depend primarily on the social group to know, who either lack ability to check for fact or refuse to check, can only be convinced by that social group they believe. And effort in their direction is simply wasted.
To me is an awesome process and definitely respect while also find it fascinating when people are told facts and believe you are trying to convince them of something.
In mathematics there is mathematical proof. And proof is definitive.
If knowledge is available, yet you believe things not true, where that can be determined by fact, then I suspect you have problems with determination of fact, or have not been informed.
With those able to access the web, who have training will admit my belief is: ability to determine fact when available represents a valuable skill.
And I also believe that some people lack ability to determine fact and simply rely on what they are told. Which is NOT necessarily a terrible thing! Most people must simply rely upon what they are told for MUCH. Like with medicine, am going to listen to that medical doctor. Probability can challenge with authority and better my own health versus endanger? Very small. Or nonexistent.
Our human reality is vastly complex with too much knowledge for any one human to have much of a grasp of it all. However, it is still possible to determine fact, if you have the skills. While some apparently require the majority to believe fact, when is determined or they may instead believe false information which to me is simply, not knowledge.
This blog audience focus from now on is to be directed towards those who can determine mathematical fact.
The ability to find knowledge should be cheered and supported. And in my analysis people who depend primarily on the social group to know, who either lack ability to check for fact or refuse to check, can only be convinced by that social group they believe. And effort in their direction is simply wasted.
Tuesday, January 16, 2018
When people work harder reducing
The burden on me to establish certainty with my own research should be vastly hard, especially with ideas that challenge the status quo. We have had a lot of great humans who have done great things, which work great too. And people flying all over with all kinds of cool technology, in countries with great freedoms, along with so much else in support of my views there. Greatness works.
So I must take my time and accept great difficulty in establishing unimpeachable certainty. Which is fun! It pushes me to understand my own research and has even aided further discovery. But also there is so much demonstrative now, done.
One of those ideas that demonstrates has to do with Diophantine equations like:
c1x2 + c2xy + c3y2 = c4 + c5x + c6y
Where is binary because you have unknowns x and y, and is also called two-variable. Diophantine just means looking for integer solution. I figured out a best way to reduce to something simple.
For example:
x2 + 2xy + 3y2 = 4 + 5x + 6y
Where I just tried something simple and was lucky there ARE integer solutions. My method for reducing gives:
(-4(x+y) + 10)2 + 2s2 = 166
And turns out you can easily solve from there and find integer solutions:
x = 4, y = -2, or x = 5, y = -2
More recently played around and had to work harder for this example, as wanting something simpler in ways:
x2 + y2 = xy + x+y + 102
Which my method reduces to give:
(-3(x+y) + 6)2 + 3s2 = 3708
And figured out solutions: x = -10, so y = -8, or x = 3, y = -8
Copying from prior posts.
Reference posts: Reducing binary quadratic Diophantines, Reducing a quadratic Diophantine to find solutions
My method for reducing is I think the best in the world, and supersedes methods that trace back to Carl Gauss who is a HUGE hero of mine. And his methods now include wasted effort finding something called a discriminant, which is not needed for reducing these equations.
People wasting their mental energy though is not surprising to me, or a great concern. Others might value their efforts more highly than I do, as is a telling failure, in my opinion.
From my perspective, is more telling when people work harder than necessary, when they could do better than those who find better.
Those who seek truth best, interest me more than those who do not.
May seem cold, but many may wish to be something they are not. To me is important how to figure out those with potential. Finding the truth when is so easily available? When easy to find on the web?
If you can't do that easy then how can you find the things waiting to be discovered, when that can be so hard?
Challenges are met by people with ability. Why I deeply appreciate the challenges in front of me, as I learn so much more about what I can do.
Everything testable and each statement of mine checkable though I wonder if am reaching in claiming is the best possible. But I think not. Like consider, how I got something important.
Use my method to reduce on: x2 - Dy2 = F
Gives the BQD Iterator.
James Harris
So I must take my time and accept great difficulty in establishing unimpeachable certainty. Which is fun! It pushes me to understand my own research and has even aided further discovery. But also there is so much demonstrative now, done.
One of those ideas that demonstrates has to do with Diophantine equations like:
c1x2 + c2xy + c3y2 = c4 + c5x + c6y
Where is binary because you have unknowns x and y, and is also called two-variable. Diophantine just means looking for integer solution. I figured out a best way to reduce to something simple.
For example:
x2 + 2xy + 3y2 = 4 + 5x + 6y
Where I just tried something simple and was lucky there ARE integer solutions. My method for reducing gives:
(-4(x+y) + 10)2 + 2s2 = 166
And turns out you can easily solve from there and find integer solutions:
x = 4, y = -2, or x = 5, y = -2
More recently played around and had to work harder for this example, as wanting something simpler in ways:
x2 + y2 = xy + x+y + 102
Which my method reduces to give:
(-3(x+y) + 6)2 + 3s2 = 3708
And figured out solutions: x = -10, so y = -8, or x = 3, y = -8
Copying from prior posts.
Reference posts: Reducing binary quadratic Diophantines, Reducing a quadratic Diophantine to find solutions
My method for reducing is I think the best in the world, and supersedes methods that trace back to Carl Gauss who is a HUGE hero of mine. And his methods now include wasted effort finding something called a discriminant, which is not needed for reducing these equations.
People wasting their mental energy though is not surprising to me, or a great concern. Others might value their efforts more highly than I do, as is a telling failure, in my opinion.
From my perspective, is more telling when people work harder than necessary, when they could do better than those who find better.
Those who seek truth best, interest me more than those who do not.
May seem cold, but many may wish to be something they are not. To me is important how to figure out those with potential. Finding the truth when is so easily available? When easy to find on the web?
If you can't do that easy then how can you find the things waiting to be discovered, when that can be so hard?
Challenges are met by people with ability. Why I deeply appreciate the challenges in front of me, as I learn so much more about what I can do.
Everything testable and each statement of mine checkable though I wonder if am reaching in claiming is the best possible. But I think not. Like consider, how I got something important.
Use my method to reduce on: x2 - Dy2 = F
Gives the BQD Iterator.
James Harris
Monday, January 15, 2018
Thoughts on infinite depth
Correct mathematics will check out correct over infinity. There is no error or possibility for error. And that is a perfect test. I like to call it--infinite depth. As incorrect mathematics on the surface can look ok, but will lead to contradiction.
Questions of whether or not mathematics can have rules that lead to contradiction were handled most famously with the work of Kurt Gödel being of signature importance. Lots of people talk him, but the essence of the desire with mathematics is what I stated.
People can though get excited over the question of: can you have valid mathematics which leads to something wrong? And I'd say simply: no.
But I see it as, if mathematics leads to something wrong, then is NOT valid. So that idea of a check possible, to me answers the question.
But how can you know? We can't check infinity with infinite tests one-by-one. Where I rely much on tautologies for my work.
Like: x+y+vz = 0(mod x+y+vz) is logically a tautology and mathematically is called an identity.
Is equivalent to: x+y+vz = x+y+vz
Much of my mathematical research reduces back to validity if that identity is valid.
Actually have a reference post from 2014 that is demonstrative:
Example, showing truth, logic and absolute proof
Identities by definition are valid; therefore, a result that so reduces is perfectly checked.
Functionally that means that a person like myself can make a mathematical statement, and find that statement implies something else. Where readily admit have come across such or had them brought to my attention and even when I know have a mathematical proof can be that emotion of FEAR.
Then test the implication and look at a new result. Discovery rules. Fear turns then to elation. But regardless, emotion is irrelevant. The math behaves perfectly. Infinite depth means there can be no mistake. And I marvel to myself or chatter a bit about it, like maybe here on this blog.
So have given the logic. Emotion can mean something else. One phrase like to use, over and over again is--math and emotion do NOT go well together. Like one way I check people is to ask, if they say absolutes do not exist, if they believe: 2+2 = 4
Number authority is so useful. With questions of truth can still be of interest to ask a person, why do you so believe? Most people will of course not debate you over such a thing, but why not?
Simple enough, yes, but there are humans who will debate you over it. I find it to be a telling human check.
James Harris
Questions of whether or not mathematics can have rules that lead to contradiction were handled most famously with the work of Kurt Gödel being of signature importance. Lots of people talk him, but the essence of the desire with mathematics is what I stated.
People can though get excited over the question of: can you have valid mathematics which leads to something wrong? And I'd say simply: no.
But I see it as, if mathematics leads to something wrong, then is NOT valid. So that idea of a check possible, to me answers the question.
But how can you know? We can't check infinity with infinite tests one-by-one. Where I rely much on tautologies for my work.
Like: x+y+vz = 0(mod x+y+vz) is logically a tautology and mathematically is called an identity.
Is equivalent to: x+y+vz = x+y+vz
Much of my mathematical research reduces back to validity if that identity is valid.
Actually have a reference post from 2014 that is demonstrative:
Example, showing truth, logic and absolute proof
Identities by definition are valid; therefore, a result that so reduces is perfectly checked.
Functionally that means that a person like myself can make a mathematical statement, and find that statement implies something else. Where readily admit have come across such or had them brought to my attention and even when I know have a mathematical proof can be that emotion of FEAR.
Then test the implication and look at a new result. Discovery rules. Fear turns then to elation. But regardless, emotion is irrelevant. The math behaves perfectly. Infinite depth means there can be no mistake. And I marvel to myself or chatter a bit about it, like maybe here on this blog.
So have given the logic. Emotion can mean something else. One phrase like to use, over and over again is--math and emotion do NOT go well together. Like one way I check people is to ask, if they say absolutes do not exist, if they believe: 2+2 = 4
Number authority is so useful. With questions of truth can still be of interest to ask a person, why do you so believe? Most people will of course not debate you over such a thing, but why not?
Simple enough, yes, but there are humans who will debate you over it. I find it to be a telling human check.
James Harris
Thursday, January 11, 2018
Some math perspective and pep talk
The math sustains me like to say, and getting going into a new year is useful to reflect on just a bit, some of how that works.
Like one of those results will use for comfort:
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.
My way to count primes in its sieve form.
That is SO compact! The math has a succinct efficiency I appreciate. One of my favorite posts talking is actually on my blog Beyond Mundane: Simple and fast prime counter
One of the last results where I got lots of criticism over a decade ago, where what I saw as specious comparison to known methods ignored the new. Yes, basic concepts were same, so?
The innovation involved is how is simply displayed and compact, which is also obvious in comparison.
Love considering as the time passes. As over 15 years now and yeah, look over what math people still teach, as if a world of students cannot also find easier on the web.
And something that so floored me only a few years ago, realizing can easily find all integer solutions for:
x2 + (m-1)y2 = mn+1
From: Squares and nth powers
Not surprisingly to me, using this result, which follows from my BQD Iterator could make for popular shares. Like:
12 + 32 = 10
82 + 62 = 102
262 + 182 = 103
282 + 962 = 104
3162 + 122 = 105
From: Sum two squares to power of 10
Simplicity has HUGE benefits as if people get curious can check you on EVERY math detail. Web enables a lot as well, as folks can know. They can just check you on the facts.
Can also check what they thought they knew, about others in a high profile area. For me was lots of surprise for years.
Now these results are comforting, while less exciting, for me.
There is that thing about the math where when you discover you get a different emotional connection to your discoveries, which I know from how I look at my math discoveries versus math from others.
The math does not care. That actually IS a comfort. Because the knowledge is more important to me.
So yeah, this pep talk is working for me like usual. And what discoveries to pick? Is just about my mood. Could talk my better than Gauss's way to reduce binary quadratic Diophantine equations. Could go on again about my modular inverse method, but am on pause there. Still relishing it though.
Or lots of other things.
Pep talk complete! Well I feel better now. Who knows why humans who come and go might fight against mathematical discovery, and do we really care?
The math feels like a friend to me. And I know the math does not care. Convincing myself, I should not either. We humans have choice.
Knowledge should be about reinforcing choice. The math has given me more choice. Am thankful.
James Harris
Like one of those results will use for comfort:
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.
My way to count primes in its sieve form.
That is SO compact! The math has a succinct efficiency I appreciate. One of my favorite posts talking is actually on my blog Beyond Mundane: Simple and fast prime counter
One of the last results where I got lots of criticism over a decade ago, where what I saw as specious comparison to known methods ignored the new. Yes, basic concepts were same, so?
The innovation involved is how is simply displayed and compact, which is also obvious in comparison.
Love considering as the time passes. As over 15 years now and yeah, look over what math people still teach, as if a world of students cannot also find easier on the web.
And something that so floored me only a few years ago, realizing can easily find all integer solutions for:
x2 + (m-1)y2 = mn+1
From: Squares and nth powers
Not surprisingly to me, using this result, which follows from my BQD Iterator could make for popular shares. Like:
12 + 32 = 10
82 + 62 = 102
262 + 182 = 103
282 + 962 = 104
3162 + 122 = 105
From: Sum two squares to power of 10
Simplicity has HUGE benefits as if people get curious can check you on EVERY math detail. Web enables a lot as well, as folks can know. They can just check you on the facts.
Can also check what they thought they knew, about others in a high profile area. For me was lots of surprise for years.
Now these results are comforting, while less exciting, for me.
There is that thing about the math where when you discover you get a different emotional connection to your discoveries, which I know from how I look at my math discoveries versus math from others.
The math does not care. That actually IS a comfort. Because the knowledge is more important to me.
So yeah, this pep talk is working for me like usual. And what discoveries to pick? Is just about my mood. Could talk my better than Gauss's way to reduce binary quadratic Diophantine equations. Could go on again about my modular inverse method, but am on pause there. Still relishing it though.
Or lots of other things.
Pep talk complete! Well I feel better now. Who knows why humans who come and go might fight against mathematical discovery, and do we really care?
The math feels like a friend to me. And I know the math does not care. Convincing myself, I should not either. We humans have choice.
Knowledge should be about reinforcing choice. The math has given me more choice. Am thankful.
James Harris
Wednesday, January 10, 2018
Power of math
That correct mathematics has a power of its own sustains me. And what I call the social problem is much about people who clearly do not believe it does.
Lies? Have to be supported constantly. That human energy can only last so long. And besides, the truth can crush lies even when LOTS of people keep trying to push them.
The valid math wins over time. And I have a unique opportunity, as NO major discoverer at my level has ever faced such opposition.
My predecessors would be jealous am sure. But they're long dead. Gives me different perspective on their lives as well. They had to fight their own battles too. History tends to be glossed over as to full reality. Still realize how much better it is, here am thankful. I get to know so much more than they ever could in ways, I think. Yeah it is different between understanding possibility and living in this modern reality.
Gist of it is, opportunity like no other as in challenges, I simply learn more useful.
From my best perspective is simply an interesting exercise.
___JSH
Lies? Have to be supported constantly. That human energy can only last so long. And besides, the truth can crush lies even when LOTS of people keep trying to push them.
The valid math wins over time. And I have a unique opportunity, as NO major discoverer at my level has ever faced such opposition.
My predecessors would be jealous am sure. But they're long dead. Gives me different perspective on their lives as well. They had to fight their own battles too. History tends to be glossed over as to full reality. Still realize how much better it is, here am thankful. I get to know so much more than they ever could in ways, I think. Yeah it is different between understanding possibility and living in this modern reality.
Gist of it is, opportunity like no other as in challenges, I simply learn more useful.
From my best perspective is simply an interesting exercise.
___JSH
Monday, January 08, 2018
So very hard to process
Have been blessed with extraordinary skepticism of my own claims. Like I claim that a destabilizing problem entered the mathematical field in the late 1800's, and somehow escaped being handled until now. To me is too wacky. And readily admit, have struggled to believe it.
Took over 13 years for me to work at convincing myself completely, while otherwise focusing on best process. So am like, well I DID get a paper published, even if things got weird after. But problem does explain that, I guess. And have other discoveries as well.
That bugged me for a bit, how long it has taken. But also realized is how gained quite a bit of functional knowledge and could test, test, test things relentlessly.
With very difficult things can just take some time.
Such a tedious thing to unravel--human error on such scale. Such a difficult thing to believe.
And HUGE was finding this latest modular inverse method May of last year, as was so clean, and so obvious--after found. That let me also consider that excitement I felt with such a discovery, as a human being, against how others behaved. Where yeah was fun to talk that out.
Ultimately the human species gained a third primary way to do a modular inverse.
Am actually, importantly, also simply the messenger.
One problem though has been trolls, where explain trolls my own way in this post and these are people who believe attention IS the point. They find it hard to process when truth is the point.
Yet they can be useful regardless.
Can you imagine? Having absolute mathematical truth and watching people do gyrations trying to talk against it? Is fun.
The social problem is hard but solvable, where yeah, in our times, expertise is so important. And maximum skepticism is required when a person is making remarkable and challenging claims about one of the most important areas of human intellectual endeavor.
For me also must admit the idea of a job aspect to it is one do not like to trot out there. Yeah to some web troll must just seem like fun to have endless global attention.
Reality? Is work.
To help humanity? I can do much. Or at least try.
Can flip so fast. Convinced then wondering, how is this situation possible?
Must work harder. There at least I have control.
It is interesting how we believe. And definitely one of the great things about my situation, without a doubt, has pushed me to better understand.
James Harris
Took over 13 years for me to work at convincing myself completely, while otherwise focusing on best process. So am like, well I DID get a paper published, even if things got weird after. But problem does explain that, I guess. And have other discoveries as well.
That bugged me for a bit, how long it has taken. But also realized is how gained quite a bit of functional knowledge and could test, test, test things relentlessly.
With very difficult things can just take some time.
Such a tedious thing to unravel--human error on such scale. Such a difficult thing to believe.
And HUGE was finding this latest modular inverse method May of last year, as was so clean, and so obvious--after found. That let me also consider that excitement I felt with such a discovery, as a human being, against how others behaved. Where yeah was fun to talk that out.
Ultimately the human species gained a third primary way to do a modular inverse.
Am actually, importantly, also simply the messenger.
One problem though has been trolls, where explain trolls my own way in this post and these are people who believe attention IS the point. They find it hard to process when truth is the point.
Yet they can be useful regardless.
Can you imagine? Having absolute mathematical truth and watching people do gyrations trying to talk against it? Is fun.
The social problem is hard but solvable, where yeah, in our times, expertise is so important. And maximum skepticism is required when a person is making remarkable and challenging claims about one of the most important areas of human intellectual endeavor.
For me also must admit the idea of a job aspect to it is one do not like to trot out there. Yeah to some web troll must just seem like fun to have endless global attention.
Reality? Is work.
To help humanity? I can do much. Or at least try.
Can flip so fast. Convinced then wondering, how is this situation possible?
Must work harder. There at least I have control.
It is interesting how we believe. And definitely one of the great things about my situation, without a doubt, has pushed me to better understand.
James Harris
Friday, January 05, 2018
More overview on modular inverse
Still giddy over finding my own primary way to calculate the modular inverse thought would talk more in general about the subject.
The modular inverse is simply the number you can multiply times another number to get 1 modulo some other number. Like 2 times 3 equals 6 which is 1 modulo 5.
Succinctly: 2(3) = 1 mod 5
Can also express as: 2 = 3-1 mod 5
Does not always exist though. Like, when prime factors are shared cannot exist. For example, consider 21 mod 33. There is no modular inverse for 21 modulo 33, but divide off the 3, and get: 7 mod 11. And modular inverse is 8, as 7(8) = 1 mod 11.
Using very simple on purpose and also shows, is not complicated. Modular inverse is just a very basic idea in modular algebra, but how do you find?
Fermat and Euler show a way
One cool way to figure it out, goes back to ideas from Fermat, though Euler expanded on. As Fermat is generally credited with noticing that a number raised to a prime power equaled itself modulo that power.
For example if x is that number, and p is the prime: xp = x mod p
What does that have to do with modular inverse?
Well you can divide both sides by x2, and get: xp-2 = x-1 mod p
For example, consider x = 3, and p = 7. Then 35 = 3-1 mod 7. Where we can take that in modular bites, as:
(32)(32)(3) = 3-1 mod 7 = (9)(9)(3) = (2)(2)(3) = 12 mod 7.
Which is 5 mod 7. And yup, 3(5) = 1 mod 7. And Euler got clever and figured out a way to use something other than a prime for the modulus using what is called his totient function. And if wish to pursue further should web search as just a bit too complicated for a quick demonstration here.
Also if don't know already should web search to learn more about Fermat's approach and why is true!
Extended Euclidean algorithm
Other way to calculate modular inverse goes back to ideas from Euclid, with something called the extended Euclidean algorithm. Gist of it is, Euclid taught a way to get the greatest common factor between two numbers. Like between 12 and 15 that number is 3. Turns out the basic approach can also be used with modular inverse. Here is a simple example, calculating modular inverse of 13 modulo 137.
Notice that: 137 = 13(10) + 7 and also that: 7*2 - 1 = 13
So: 7*2 = 1 + 13, and: -13 + 7*2 = 1
Looks weird am sure, but I want that 1, and can substitute now for 7, from first as:
7 = 137 - 13(10), so: -13 + 2(137 - 13(10)) = 1
Since I want modulo 137 can make things easier: -13 - 2(13)(10) = 1 mod 137
And simplify to get: -13 - 13(20) mod 137 = 13(-21) = 13(116) = 1 mod 137
So, 13(116) = 1 is found, and notice just did a few things kind of different, but clever. But unlike before there is no way to know how long that will take to work!
Turns out, key to the quick there is that 137 mod 13 = 7 and 7*2 = 1 mod 13. So was lucky. But you WILL get to something like that eventually with this approach, if no prime factors are shared, otherwise you get to all shared prime factors. You have to find that situation where you get a 1, to find your modular inverse. And if is new to you, and wonder how that works, which is rest of the trick, yup, web search. Am just doing simple and quick here to give an overview of the basic math principles involved.
My way is latest primary way
Besides just going through residues and multiplying out checking which one is modular inverse, which is called using brute force, the two basic approaches above were ALL the primary ones that humanity had. And lots of variations on those themes too as clever people figured out how to do various ways.
Yet regardless, each method for calculating modular inverse traced back to only the two basic or brute force.
Eight months ago though, on May 5th, 2017, introduced the first new basic approach in a LONG time, with a third primary way, and am celebrating that a bit with this post! One of my best explanations I think is:
My modular inverse method
And I've talked a LOT around how I found it, and why that is surprising, as it is.
This post was focused on the other two primary ways for finding the modular inverse.
James Harris
The modular inverse is simply the number you can multiply times another number to get 1 modulo some other number. Like 2 times 3 equals 6 which is 1 modulo 5.
Succinctly: 2(3) = 1 mod 5
Can also express as: 2 = 3-1 mod 5
Does not always exist though. Like, when prime factors are shared cannot exist. For example, consider 21 mod 33. There is no modular inverse for 21 modulo 33, but divide off the 3, and get: 7 mod 11. And modular inverse is 8, as 7(8) = 1 mod 11.
Using very simple on purpose and also shows, is not complicated. Modular inverse is just a very basic idea in modular algebra, but how do you find?
Fermat and Euler show a way
One cool way to figure it out, goes back to ideas from Fermat, though Euler expanded on. As Fermat is generally credited with noticing that a number raised to a prime power equaled itself modulo that power.
For example if x is that number, and p is the prime: xp = x mod p
What does that have to do with modular inverse?
Well you can divide both sides by x2, and get: xp-2 = x-1 mod p
For example, consider x = 3, and p = 7. Then 35 = 3-1 mod 7. Where we can take that in modular bites, as:
(32)(32)(3) = 3-1 mod 7 = (9)(9)(3) = (2)(2)(3) = 12 mod 7.
Which is 5 mod 7. And yup, 3(5) = 1 mod 7. And Euler got clever and figured out a way to use something other than a prime for the modulus using what is called his totient function. And if wish to pursue further should web search as just a bit too complicated for a quick demonstration here.
Also if don't know already should web search to learn more about Fermat's approach and why is true!
Extended Euclidean algorithm
Other way to calculate modular inverse goes back to ideas from Euclid, with something called the extended Euclidean algorithm. Gist of it is, Euclid taught a way to get the greatest common factor between two numbers. Like between 12 and 15 that number is 3. Turns out the basic approach can also be used with modular inverse. Here is a simple example, calculating modular inverse of 13 modulo 137.
Notice that: 137 = 13(10) + 7 and also that: 7*2 - 1 = 13
So: 7*2 = 1 + 13, and: -13 + 7*2 = 1
Looks weird am sure, but I want that 1, and can substitute now for 7, from first as:
7 = 137 - 13(10), so: -13 + 2(137 - 13(10)) = 1
Since I want modulo 137 can make things easier: -13 - 2(13)(10) = 1 mod 137
And simplify to get: -13 - 13(20) mod 137 = 13(-21) = 13(116) = 1 mod 137
So, 13(116) = 1 is found, and notice just did a few things kind of different, but clever. But unlike before there is no way to know how long that will take to work!
Turns out, key to the quick there is that 137 mod 13 = 7 and 7*2 = 1 mod 13. So was lucky. But you WILL get to something like that eventually with this approach, if no prime factors are shared, otherwise you get to all shared prime factors. You have to find that situation where you get a 1, to find your modular inverse. And if is new to you, and wonder how that works, which is rest of the trick, yup, web search. Am just doing simple and quick here to give an overview of the basic math principles involved.
My way is latest primary way
Besides just going through residues and multiplying out checking which one is modular inverse, which is called using brute force, the two basic approaches above were ALL the primary ones that humanity had. And lots of variations on those themes too as clever people figured out how to do various ways.
Yet regardless, each method for calculating modular inverse traced back to only the two basic or brute force.
Eight months ago though, on May 5th, 2017, introduced the first new basic approach in a LONG time, with a third primary way, and am celebrating that a bit with this post! One of my best explanations I think is:
My modular inverse method
And I've talked a LOT around how I found it, and why that is surprising, as it is.
This post was focused on the other two primary ways for finding the modular inverse.
James Harris
Thursday, January 04, 2018
Sharing knowledge is fun
Recently discovered could share to a math group on Reddit, which has been interesting. And experience with sharing there is also useful for talking trolls! To me, sharing something you think is interesting is not a big deal. However there can be people who get upset by it, or wish to exploit for attention to themselves, which interests me.
Here's a link to the discussion: Our species can miss obvious major math till we don't, but why?
Remarkably MOST of my comments have been suppressed. Check it out. Fascinating, eh? People there did it quite deliberately too.
SOME people seemed to be interested, but they were quickly drowned out by trolls. If you're curious about troll behavior, check out that link. Is on full display.
And word troll in web context comes from phrase--trolling for attention. Since they apparently lack ability to draw attention on substance, or are unwilling to rely upon, they try to steal at the expense of others. People trolling for attention use a variety of trolling behaviors. Often they rely on insults of some kind. Hurting a target is a way to get a person to engage, which trolls seek to, yup, draw even more attention.
Well at least there WERE some people who seemed generally interested. Is very hard to contain trolls. They are the attention parasites of the web. Learned much though! Am glad found some place else to post math.
Reddit amuses me.
So what is the purpose in sharing?
Is interesting to me how often seems convoluted in our times.
Oh yeah, Reddit folks if see this message, feel free to comment here as well! Yes, I moderate here, but try my best to not interfere with opinions. As long as people are not just being rude or deliberately disruptive.
And yes, am aware someone on Reddit decided to insult me with a post saying is bad math. (If you didn't notice on bad math board, you didn't miss anything.) And use smears as evidence. No math presented, and no refutation of my math. Just insults. Some people feel ownership in areas where none exists.
Smart readers can check for themselves as to truth or gratuitous insult, but why? Is about THAT person, and not about me or my math actually.
Math does not need such emotion. People demonstrate that anger, with actions, like insults meant to hurt.
And see misinformation spread, like I do NOT work on Fermat's Last Theorem as asserted on the bad math board. Ancient history as DID work at long ago, with spectacular failures, but abandoned such efforts. Yet their system means my comments noting? Suppressed. So curious. Reddit seems to allow a massive amount of censorship. Let's see how well that works, eh? I find it challenging. Who wins this one.
Still has cool things. Like if you wish to see comments they hated so much? Here you go:
My Reddit comments
That is a great feature, can share your comments yourself! Suggest readers look around there. Reddit has made strides in trying to get to decent conversation, but as I observe am seeing that they STILL have people who lash out more than reason, and get away with it too.
Reddit does try I think. But is SO hard to keep humans from just being mean and nasty. I can see they have lots of problems in that area. But still at least they are trying, I think. Is a difficult problem.
Like how can link to things though so people can see for themselves. Like see what comments could supposedly be so horrible as to lower what they call karma. I think instead you will see inside of Reddit, like with an x-ray as to how things actually are. Fantasy is one thing, but facts rule.
Some people can earn attention, but others try to steal it.
Feel free to tell me YOUR opinion with a comment.
My guess though is that most will stick with where commenting is comfortable which is ok.
Obviously got a lot of response to this blog which is why am addressing with a post. Thanks Reddit! My goal is simply to share interesting math, because I like it.
Is fun to share.
Update Jan 10: Decided to do a follow-up posted yesterday to narrow to more useful discussion. No responses AT ALL. My conclusion is that Reddit focuses people on sharing to get attention. Which doesn't interest me. I think sharing should be to share useful information.
Link to discussion area: Overview on modular inverse
Am still looking with curiosity over what happens. It is possible my last posting was significantly--and invisibly to me as they seem to get a thrill out of that--censored by Reddit's systems. That is worth further investigation, in and of itself.
Obviously, I can work around such simple censorship by posting here, as long as Reddit doesn't start bocking that ability too! Is fascinating to me how much censorship ability is built into Reddit. Fascinating.
Well I don't blame them for that. You can get a lot of problems with hostiles. And was investigating as Reddit comes up a lot to see what progress they had made. My assessment so far?
They've simply employed a system of massive censorship capability with a chosen group who are the censoring agents, and congratulated themselves. Is not that easy though, as people abuse such power.
They need to study human history.
James Harris
Here's a link to the discussion: Our species can miss obvious major math till we don't, but why?
Remarkably MOST of my comments have been suppressed. Check it out. Fascinating, eh? People there did it quite deliberately too.
SOME people seemed to be interested, but they were quickly drowned out by trolls. If you're curious about troll behavior, check out that link. Is on full display.
And word troll in web context comes from phrase--trolling for attention. Since they apparently lack ability to draw attention on substance, or are unwilling to rely upon, they try to steal at the expense of others. People trolling for attention use a variety of trolling behaviors. Often they rely on insults of some kind. Hurting a target is a way to get a person to engage, which trolls seek to, yup, draw even more attention.
Well at least there WERE some people who seemed generally interested. Is very hard to contain trolls. They are the attention parasites of the web. Learned much though! Am glad found some place else to post math.
Reddit amuses me.
So what is the purpose in sharing?
Is interesting to me how often seems convoluted in our times.
Oh yeah, Reddit folks if see this message, feel free to comment here as well! Yes, I moderate here, but try my best to not interfere with opinions. As long as people are not just being rude or deliberately disruptive.
And yes, am aware someone on Reddit decided to insult me with a post saying is bad math. (If you didn't notice on bad math board, you didn't miss anything.) And use smears as evidence. No math presented, and no refutation of my math. Just insults. Some people feel ownership in areas where none exists.
Smart readers can check for themselves as to truth or gratuitous insult, but why? Is about THAT person, and not about me or my math actually.
Math does not need such emotion. People demonstrate that anger, with actions, like insults meant to hurt.
And see misinformation spread, like I do NOT work on Fermat's Last Theorem as asserted on the bad math board. Ancient history as DID work at long ago, with spectacular failures, but abandoned such efforts. Yet their system means my comments noting? Suppressed. So curious. Reddit seems to allow a massive amount of censorship. Let's see how well that works, eh? I find it challenging. Who wins this one.
Still has cool things. Like if you wish to see comments they hated so much? Here you go:
My Reddit comments
That is a great feature, can share your comments yourself! Suggest readers look around there. Reddit has made strides in trying to get to decent conversation, but as I observe am seeing that they STILL have people who lash out more than reason, and get away with it too.
Reddit does try I think. But is SO hard to keep humans from just being mean and nasty. I can see they have lots of problems in that area. But still at least they are trying, I think. Is a difficult problem.
Like how can link to things though so people can see for themselves. Like see what comments could supposedly be so horrible as to lower what they call karma. I think instead you will see inside of Reddit, like with an x-ray as to how things actually are. Fantasy is one thing, but facts rule.
Some people can earn attention, but others try to steal it.
Feel free to tell me YOUR opinion with a comment.
My guess though is that most will stick with where commenting is comfortable which is ok.
Obviously got a lot of response to this blog which is why am addressing with a post. Thanks Reddit! My goal is simply to share interesting math, because I like it.
Is fun to share.
Update Jan 10: Decided to do a follow-up posted yesterday to narrow to more useful discussion. No responses AT ALL. My conclusion is that Reddit focuses people on sharing to get attention. Which doesn't interest me. I think sharing should be to share useful information.
Link to discussion area: Overview on modular inverse
Am still looking with curiosity over what happens. It is possible my last posting was significantly--and invisibly to me as they seem to get a thrill out of that--censored by Reddit's systems. That is worth further investigation, in and of itself.
Obviously, I can work around such simple censorship by posting here, as long as Reddit doesn't start bocking that ability too! Is fascinating to me how much censorship ability is built into Reddit. Fascinating.
Well I don't blame them for that. You can get a lot of problems with hostiles. And was investigating as Reddit comes up a lot to see what progress they had made. My assessment so far?
They've simply employed a system of massive censorship capability with a chosen group who are the censoring agents, and congratulated themselves. Is not that easy though, as people abuse such power.
They need to study human history.
James Harris
Tuesday, January 02, 2018
Why math rules
There is a greatness to certainty from absolute proof. And with my story appreciate how fascinating can be when your fellow human beings wish to argue against it. Mathematics allows you to watch that behavior calmly.
Why do people do it?
People can argue against absolute mathematics primarily in my opinion because it feels good. People can cherish that false sense of control, in belief, even when it is provably wrong.
Truth always wins, one way or another, but humans can convince themselves otherwise. Is an odd characteristic of our species.
I like to study that will admit. I find it curious more and more, as watch.
However, the math does not change. And over time, feelings subside. The human animal will only expend so much emotion. Or those people go, and new ones are born.
Valid math does not change. The math does not care.
Valid mathematics can withstand any challenge.
Mathematics rules over time, always.
The math does not notice.
The discoverers get to have SO much fun. Am learning to really appreciate and enjoy it.
James Harris
Why do people do it?
People can argue against absolute mathematics primarily in my opinion because it feels good. People can cherish that false sense of control, in belief, even when it is provably wrong.
Truth always wins, one way or another, but humans can convince themselves otherwise. Is an odd characteristic of our species.
I like to study that will admit. I find it curious more and more, as watch.
However, the math does not change. And over time, feelings subside. The human animal will only expend so much emotion. Or those people go, and new ones are born.
Valid math does not change. The math does not care.
Valid mathematics can withstand any challenge.
Mathematics rules over time, always.
The math does not notice.
The discoverers get to have SO much fun. Am learning to really appreciate and enjoy it.
James Harris
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