## 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.

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 that every power of 10 can be written 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

## Saturday, August 12, 2017

### Some number examples

For some reason really like when have examples with numbers in posts and some are actually my favorites and thought to collect a few linking back to post, in a post.

Like here is one:

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

615+ 30752 + 141452 + 159902  + 1884972   =  774*210

Just like to stare at it for some reason.

Seem to like sums of lots of squares. Continuing, like here is two more sums of 5 squares to a square:

42 + 6+ 10+ 14862 = 882

and

862 + 129+ 215+ 301+ 8812 = 9682

And here went ahead and summed 7 to get a square:

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

Where am using BQD Iterator for all of these. But is mathematical tool I have which makes such things easy. Those are ones just from my own research.

An example from even earlier though, where rely on previous known result is here where was talking size of what I now call the unary form of the two conics equation:

60*2551100302 + 2551100292 = 19924730292

That is related to something, talk it here and for reference: 297182 - 61*38052 = -1

Liking that the easier solution, which is historically known so know from other sources and didn't figure it out myself, fits nicely there.

Putting in one place is useful to me for staring at them purposes.

James Harris

## Friday, August 11, 2017

### Some math discoveries listed

Sometime this month, as don't remember exact day, will be 15 years since found my prime counting function. And thought along with that special event would just list some of my mathematical discoveries. Will not link to anything as everything is on blog somewhere.

So first, yeah found my own prime counting function fastest for its size in its compact sieve form, but more importantly leads to a difference equation when fully mathematized, which has to be constrained to get it to count primes. But THAT difference equation importantly leads to a partial differential equation.

So world was given a clear and direct explanation for how prime counts can connect to continuous functions.

That is so cool. Is so weird though, how close some were to the simple explanation before, without finding it. Their methods for counting primes were SO close. Oh well, left for me to find.

Too much emotion in area explains delay from official figures acknowledging.

Emotion and mathematics? Do not go well together.

Should I admit a sort of grim satisfaction whenever go looking for current research on Riemann Hypothesis? Hardly matters for me, of course. Yeah, like I said, emotion and mathematics do not go well together.

And celebrate 15 years this month since first found my prime counting function.

Moving on.

Developed my own mathematical discipline using what I call tautological spaces, as rely on complex identities.

Entire field call modular symbology, and it realizes the first true modular algebra.

And is my best example of my use of abstract reductionism. Oh, before called modular algebra symbology, but like it shorter! So will switch to that now. There is so much related it dominates this blog.

Oh yeah, so in that area found my own way to solve for the modular inverse!

That is FINALLY for the human species a third way, with other two there is Euclid's name on one and Euler's name on the other. Is now barely over three months since found that so still absorbing the thrill.

And found an axiom related to primes, decided to call prime residue axiom.

Biggest thing though, is found my own numbers. That one takes SO much explaining will leave it like that, but big part was finding my own ring which call the object ring.

And that I think covers enough of the highlights to satisfy my mood.

James Harris

## Saturday, August 05, 2017

### Web rules and my Diophantine reducer

One of my more informative results can help elucidate how the web has changed things with sharing even highly refined information, like consider if wanted to reduce:

x2 + 2xy + 3y2 = 4 + 5x + 6y

Can use my method for reducing to get: [-4(x+y) + 10]2 + 2s2 = 166

Which is an example have used since 2011, where s = 9, and x+y = 2, is one solution, from which you can find: x = 4, y = -2 as one solution. And another gives: x = 5, y = - 2

Kind of cool, huh? Same y works for two different values for x. And really glad the equation had integer solutions! And easy ones too. Is obvious why I picked it am sure. And s has an explicit solution as a function of x and y, but I don't use it, as just helps by giving two linear equations to solve for x and y, but DO talk about it in this post.

And copied example from this post showing my way to reduce what are called binary quadratic Diophantine equations and have also seen called two variable quadratic Diophantine equations.

For me reducing is for show.

But there are people who need to reduce these types of equations for am sure lots of reasons, so linking to my method is about usefulness. However, why link to something worse than other techniques? To do so would be illogical, and against common sense.

In fact have noted my method improves upon methods for reducing learned are from Gauss. But with his techniques you also need to check something called a discriminant. With my approach that is worthless effort, and I make no mention of such a thing.

Turns out you don't need it.

But what do you think is in some math textbook, eh? And am a HUGE fan of Gauss, but if he were alive today doubt he'd be surprised that his authority is not taken lightly. Who knows when academics will update.

Innovation tends to lead in dramatic ways which is more fun and rapid. People who NEED will just go to the best thing available, when know it is. Getting established? Is more of a process which to me is tedious and depends on others, whose motivations can vary.

Web makes all that irrelevant from MY perspective.

Web can just connect information DIRECTLY to the people who need it.

With such older research of mine the web is very efficient in linking to it. And I can check search engines based on my own results to see if search results will get to them, and do it routinely.

Usually Google wins, and will check against Bing more than others. However, at times with more recent research have seen Bing win. And I think Google is making more effort to rely on establishment authority, rather than just on web authority, which is a hypothesis to explain that result.

Over time though, best results will win, and notice that is true regardless of the web.

And academics lagging best methods is not new I don't think. The web though can simply link to best, though I think often is being done on various views of authority! Which I think is interesting.

Picked one of my most dramatic examples but can help explain other areas too. Right now have several results could have used, but I like this one as is connected to some serious practical things.

World doesn't sit and wait on academics. And never has. But web has made things easier.

So yeah, one thing I do routinely is check search engines to see better how they operate based on what happens with my own research results, when can check those objectively against what is known, where usually can.

Figuring out web rules is of interest to me.

James Harris

## Wednesday, August 02, 2017

### Prime difference and marking fifteen years

Things have worked out really well for me from a discovery perspective. And it is important should note this month will mark 15 years since found my prime counting function, which in its most mathematized form counts using a difference equation. I think at times have used the word mathematicized. Did I make that up? Guess so, as spell check doesn't like it.

And don't know when this month though so may as well make a post now. Just remember was August 2002, and guess could try to check? But is some post on Usenet and do NOT wish to dig through there.

On here as checking published posts now, see this post as first one, which was posted June 2005. And first post on blog is March 2005, so took a few months to get to it. Of course this was second thing I had for my math? Before had something before Blogger and was paying for a website so was very happy when could switch to something free. That is, talking webpages. For a LONG time was just arguing on Usenet and eventually through Google Groups at some point. I say a long time but think was from 1996?

There are SO many posts on this blog talking my prime counting function where think have labeled most, so if curious can go through.

That prime counting function was first result which really was just deriving something kind of because felt like it. Specifically asked myself to figure something out and few weeks later had it, really just from scratch.

My prime counting function is also my most stand-alone result. It doesn't derive from any other research I'd done, and it is very compartmentalized though can get sort of broad. So there is a sieve form, and a fully mathematical form, where yeah guess that is supposed to be called fully mathematized. And there is the difference equation itself which has to be constrained for it to give the count of prime numbers. Then there is the partial differential that follows from the difference equation. And that is all really just one package all to itself.

Was also first result where there was just no doubt as wasn't using techniques I'd invented or wondering how something so simple could be missed. Oh, was a bit before I had the partial differential equation that follows, but had it by the time created this blog under its old name. And that is this post, which was also June 2005. Updated that post at some point to use delta symbol.

So cool. Has been SO amazing on the discovery side. Changed my life for sure.

James Harris

## Tuesday, August 01, 2017

### Sum two squares to power of 10

Number authority is from the knowledge embedded in numbers which is infinite and absolute. Like consider:

12 + 32 = 10

82 + 62 = 102

262 + 182 = 103

282 + 962 = 104

3162 + 122 = 105

So yeah, every power of 10 out to infinity is a sum of two squares, which doesn't seem useful to me, just curious. What can it tell me? Such things I enjoy pondering though yeah, also just like looking at numbers as you may have noticed.

Am just using a specific of a more general result as 10 is such a big deal to many, including me.

The numbers KNOW and we can find things out, if we want. So much depends though, on asking questions. The numbers know infinite information. We can find some things out.

If it doesn't interest you, ok.

For me such things DO interest, and this result follows from a simple rule.

If:  u2 + 9v2 = 10a

Then: (u - 9v)2 + 9(u + v)2 = 10a+1

Which is just using my BQD Iterator. And notice that u and v can be positive or negative, while I like to show positive as is easier and looks prettier. Copied much from an earlier post where you can go there to read more.

And there are LOTS of posts on my BQD Iterator, and you can find a list of those I've labeled by clicking on label beneath this post.

## Monday, July 24, 2017

### When truth is your hammer

Readily admit I will turn to absolute proof and even number authority itself, when need reassurance. And when you wield truth in a certain way it can be necessary to hammer through against people who rely more on feeling than fact, as we humans can be recalcitrant at times. And there are people who think truth is a moving target or most amazing to me, some think truth is about human opinion.

So yeah I posted this thing recently:

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

_____________________________

An absolute truth it doesn't care what you think of it, or me.

And I'd summed five squares before and the math is easy, but it just felt good. And I like to stare at it when get philosophical or, yeah need that reassurance.

Human beings will come and go.

Here in this time, when some may think that they have a will that can overcome mathematical proof.

Let them try.

When you are someone like me who wields truth as a hammer when necessary, or has the infinity results, then you can look at others with curiosity, to see what they believe.

Try to break the hammer, see what happens.

I'm curious. I, at least, am human. But in the end?

The math is not human. The math knows you. But you do not control the math.

The math does not care what you believe.

Possibly some of you as well will feel that urge to wield truth.

Do you have what it takes? Few can handle the truth at certain levels, who can find the most powerful infinity results and present to their world.

I could. I have.

This post was SO much fun. Yeah am a fan of comic books. But I read them a little differently than others am sure, now. And the movies? Are so much more fun, for me.

Takes someone like me.

Can you stand with truth as your best protector?

Or would you bend in fear?

I know.

For me maybe is more fun this way anyway, as suits my flair for the dramatic.

Coming up on 15 years since I discovered my prime counting function as just one example. Can you imagine? Could you simply stand for truth?

Challenge demands a certain person. Reality? Knows.

The math knows. The math chose. Reality bends not for you.

The future demands the one who will get it done.

There are more results out there--an infinity of them. Reality will choose those who will get it done.

Can you stand for truth? Come what may?

Readily admit as much as I LOVE discovery of my own, more and more wonder, where can these ideas lead that others might find?

Truth is out there.

James Harris

## Thursday, July 20, 2017

### Progression, abstraction and two conics equation

Worth noting the progression to my latest result, which has a lot to do with reducing the general equation for what I like to call a binary quadratic Diophantine equation which is also called a two variable one:

c1x2 + c2xy + c3y2 = c4 + c5x + c6y

Here x and y are the two unknowns to be figured out. The base result comes from this post from September 2008. But checking published posts, was referenced talking a general method for reducing binary quadratic Diophantine equations on this blog in this post in May 2011.

And copying from this post, where also note that with my method for reducing can get to the general reduced form:

u2 - Dv2 = C

Where u and v are unknowns. And while I've talked about with C=1 as the two conics equation before, the more general also gives two so that is just the unary case. Letters don't matter of course and like to show as:

x2 - Dy2 = F

where all variables are non-zero integers. And yeah a LOT of abstraction, in that progression, where now you can solve for x and y modularly. Looks like I figured that out in September of 2012.

With a non-zero integer N for which a residue m exists where--m2 = D mod N, and r, any residue modulo N for which Fr-1 mod N exists then a solution is:

2x = r + Fr-1 mod N and 2my = Fr-1 - r mod N

And use that to solve for the modular inverse.

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

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

And felt an urge to put all together to kind of see that progression from the most general form, to abstraction to the more basic two conics form, and then to a solution for the modular inverse.

So you end up going from something to do with binary quadratic Diophantine equations to something more general than them.

James Harris

## Wednesday, July 19, 2017

### Correct matters most

When found mathematical results have an absolute aspect you can find difficult to find elsewhere, like maybe only in logic. When you have the correct mathematical result, it is absolutely true. And have given an example of absolute proof.

However emotion can lead us astray as human beings and have been lead astray in the past by my emotions and did not like it when found out! Where could be SO confident and certain, when thought had a correct mathematical argument only to finally have that wrong belief dramatically collapse when finally could see my error.

That elicits a terrible feeling and I do not like it. I try not to repeat such failure.

The joy in believing you have something important is not worth it, if it is not even correct, as such a thing is completely empty.

Correct matters most, as only when absolutely correct do you have the mathematical proof, as proof is perfect.

To check against emotion I now employ a process which I think helps protects others as well, as while not good to lead one's self astray, so much worse to lead others! Which means I try to focus objectively, consider results as facts only when well established, and refrain from emotional appeals.

So please do not be surprised at a steady process which does not involve trying to convince you, but is sharing of mathematical ideas and process as well, so that truth can be determined.

For those who appreciate truth, working for the truth should be a privilege.

Am lucky in that most of what I have requires only what are generally called elementary methods. I like that phrase. Elementary methods.

Numbers have fascinated me in special ways for as long as I can remember. Like friends with personalities who are anxious to tell you cool things. And they never lie. But can lie to myself if I'm not careful so yeah, focus on--correct matters most.

And explanation helps.

Labels below this post consider various areas around the social aspect of presenting mathematics from celebrity, to what I call the social problem, and also instructional. Click on a label for more posts in that area! And thank you for your interest.

I try not to try to convince you, but I do appreciate your time and attention.

James Harris

## Monday, June 12, 2017

### Number authority

Find that I DO turn to numbers routinely. And the authority you feel when the numbers behave as mathematics requires is like no other to me.

The math does not care. But the math is never wrong.

And there is a comfort in that which I think turns into a sense of protection. So the math may not care but you can find shelter in truth.

So yes, will turn to numbers in comfort and love the conversation with the math even knowing the math does not care. But the math can talk to you. And you can talk to the math, and ask questions!

And if you ask the right questions, the math can give you the truth.

Without a doubt to me is one of the greatest phrases. And in mathematics truth can be found--without a doubt.

James Harris

### Summing seven squares to a square

Discovered a simple technique to build sums of as many squares as you want to a square. For example, here is a sum of seven squares to get a square:

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

_____________________________

My basic result used to get an arbitrary length of sums of squares is that in general there must always exist nonzero x and y, such that for an integer n equal to 1 or higher, and an integer m equal to 3 or higher:

x2 + (m-1)y2 = mn

Where n starts at 1. Often I like to start it at zero so it's a count of iterations but is prettier starting at 1. And you get values for x and y using what I like to call a BQD Iterator, which is short for binary quadratic Diophantine iterator. I've talked about it a lot.

For a sum of c+1 squares: m = s12 +...+sc2 + 1

x2 + (s12 +...+sc2)y2 = mn

The BQD Iterator is:

Given nonzero integers u and v with

u2 + (s12 +...+ sc2)v2 = F

then it must also be true that

(u - (s12 +..+sc2)v)2 + (s12 +...+sc2)(u + v)2 = (s12 +...+ sc2 + 1)*F

So for 7 squares, I'll need 6 s's and I'll use primes: 2, 3, 5, 7, 11 and 13

Then m = 4 + 9 + 25 + 49 + 121 + 169 + 1 = 378

12 + 377*12 = 378

First iteration: (-376)2 + 377*(2)2 = 3782

Second iteration: (-1130)2 + 377*(-374)2 = 3783

Third iteration: (139868)2 + 377*(-1504)2 = 3784

Which is: (139868)2 + 4*(-1504)2 + 9*(-1504)+ 25*(-1504)+ 49*(-1504)2 + 121*(-1504)2 + 169*(-1504)2 = 3784

Can divide both sides by 16, and get rid of negatives to get:

Which is: 349672 + 7522 + 9*(376)+ 25*(376)+ 49*(376)2 + 121*(376)2 + 169*(376)2 = 1894

And now get final result where will show as all squares:

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

Which is interesting to me, I think. Looks more impressive that way. Of course watch it get built maybe less impressive? But still is, a sum of seven squares to get a square.

Don't really see a practical use, so to me? Is just pure math.

Am curious, if you know a number theorist, why not ask that person to produce an example of sums of squares to a square? It's not like it's actually hard to do, if you know how.

Of course, highlighting a cool result with mathematical tools I pioneered. Do I really know or care if number theorists can match me here? Not really.

For the discoverer? It's all good.

James Harris

## Friday, June 09, 2017

### Talking my modular inverse discovery

Figured out my own way to calculate the modular inverse, adding now a third primary way, where before there were only two.

The modular inverse is a rather simple thing from modular arithmetic, like consider:

2(3) ≡ 1 mod 5

Here 2 is the modular inverse of 3 and vice versa because they multiply to have a residue of 1 modulo 5, which is the modulus. That can be written as:

≡ 3-1 mod 5

Importantly in my research as I use SO much modular long ago tired of copying and pasting the modular congruence symbol so just use equals, so I have:

2(3) = 1 mod 5 and then: 2 = 3-1 mod 5

May seem small but in my experience you can face purists who will be dismissive on such small matters! When am someone who is NOT going to waste time copying and pasting something all over a vast amount of research just to appease such people as is human convention. The math does not care.

The modular inverse as a concept has been around for some time, but only a few basic approaches for finding it were previously known. One relies on something called the extended Euclidean algorithm, which is very simple but will just link. The other approach depends on something called Euler's theorem, where learned the above from article on Wikipedia on what they call the modular multiplicative inverse.

And now can add another basic approach which relies on a system of equations I discovered about a month ago. For some residue r modulo N, its modular inverse is:

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. And my system, is kind of more direct like that which follows from Euler's theorem, but is also iterative like what follows from extended Euclidean algorithm. But allows you to fiddle with things in a way that neither does. So you can pick two key variables: m and y0

And they are so named because of how I discovered the system, as was just kind of puzzling over some things.

Here is a link to a post showing an iterative example.

It is one of my most direct discoveries which surprised me a bit, as it just flowed. And I'm beginning to accept that I have years of experience which has given me a certain level of expertise.

Research about the modular inverse on the web I've done since, when yeah got REALLY interested, has indicated is also a practical result as calculations of the modular inverse are part of modern techniques, according to that research. So it has applied and pure math aspects.

The result is definitive though in terms of evaluating social aspects of how discovery is actually treated versus how one might imagine. I've had lots of experience in this area with prior results.

And the social problem label below covers that topic.

Good news though, looks like first basic major result at this level in over a century. Probably MUCH longer. Notice prior methods go back to Euler and Euclid. Wow. Supposedly such a possibility of such a basic find no longer existed. Am thrilled to be the discoverer.

So yeah, humanity has a new mathematical tool.

Mathematics discovered belongs to the human race.

James Harris

## Sunday, June 04, 2017

### Iterative example with my modular inverse method

My way to calculate the modular inverse at least gives a smaller modulus with another inverse to calculate and realized might help to show an iterative example.

Will start with the system which I will copy from my reference post, and then will explain lots, where am not doing so much shown calculation as is tedious. So very wordy post I warn. Less with shown work. Interested readers are invited to work through themselves as I think greatly helps understanding to fill in what I'm leaving out.

Also lets me explain some things a bit differently. 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.

Importantly you have variables that follow from: x2 - m2y2 = F

And you're solving for n and d, with other variables picked. Where I've tended to pick m such that m2 is greater than N, though is NOT clear if that is necessary. Is a new area I just found.

I do know that m should not equal r though. So let's try it all with N = 851 = 23(37).

I tend to let y0 = 1, where that is just for easy. Later research may indicate other more useful values. And will let m = 31, as that squared is greater than N and I like primes. Not necessary for m to be prime though.

Oh, and let's let r = 97. I like using primes because things won't work if r shares factors with N, of course as then the modular inverse does not exist! So helps me just using primes often.

Now can calculate F0, which is: 97(97+2(31)) mod 851.

So F0 = 105 mod 851.

Plugging everything in and reducing things and simplifying, I get:

298d = 325n - 166 mod 851

So now to solve for n can just find: 325n - 166 = 0 mod 298

That is: 27n = 166 mod 298, so now I need the modular inverse...oh wait. I notice that 298 - 166 has 3 as a factor, so can get:

9n = -44 mod 298

Oh, then I got clever. If we let n = 2z, then: 9z = -22 mod 149, as can divide 2 across everything.

So now finally, can look for modular inverse of 9, modulo 149. It works. Not in the mood to put all that here though. But just use the system already given and this time it just gives the answer without further iteration needed. Turns out 9-1 = 116 mod 149. Where just get the answer that time without needing to iterate further.

Will leave as an exercise for the reader to get to: n = -38 mod 851, d = -42 mod 851

(I think.)

Where since is modular algebra those aren't the only ones that will work. But I liked them because they're small. Also shows you can use negative as well as positive, of course.

Then r-1 = -39(159) - 62(-42) mod 851

And r-1 = 658 mod 851. And 97(658) = 1 mod 851 as required.

I think that's all ok, as to how it works. Copying from notes so may be minor errors here or there.

Main point is: you definitely get to a smaller modulus as went from 851 to 149. And also can just iterate with THIS method, so it will solve for the modular inverse.

Lots of research room though for best techniques.

And I noted some open questions, like what is best choice for m? Or y0? And does it matter to work more to pick a smaller F0? I just went with whatever. You can fiddle with things to make coefficients of n and d as small as possible, I'd think!

So much room for further research.

James Harris

## Thursday, June 01, 2017

### Solid result and am surprised

Just a quick update that my modular inverse solving method is solid. Luckily is easy math, so not much to check. For those who wonder, when an error is in there it usually is something simple. I have my own definition of mathematical proof for checking for errors, but human reality can be you just miss things. But my modular inverse innovation is perfect.

It adds to my collection of infinity results. And that label is below the post for those curious about the others. Just click on labels below posts to get other posts where I've so labeled.

But so wild was available! Am so glad just kind of was wondering and have had that modular factorization for YEARS and talked it up on this blog too. Oh yeah, decided had a responsibility to at least try and sent some emails to some mathematicians and one US Government agency. Felt like the right thing to do. No replies from the mathematicians. And just an auto-reply so far from US Government agency.

That's ok. Just doing my due diligence. But for those who wonder? No, am not expecting mathematicians to reply to me any more as what can they say? Think about it. Any reply can force them to do more than just reply, like help champion a really massively cool discovery which greatly adds to human knowledge.

Maybe I should give them the benefit of the doubt. Hasn't been that long. Still feel there is clearly indication of a certain poetic justice in play. Situation can be distressing for others am sure who may worry about controversy! But good news is, is more me disappointed than anything else. I have math ideas I'd just as soon see picked up by established mathematicians. Mostly though they just seem to ignore me.

Regardless of any of that am SO excited with the find!

Am so happy with this thing. Not my biggest result by far, but one that tells a lot. Why was it available for me?

Because I went looking for it, I guess. I don't know. Am just babbling now. Then again is also would think another example of the analytical power of abstract reductionism.

Main thing: is a solid result. May have a massive impact in many areas of number theory, remarkably enough. I suspect is ALREADY being used by now as information travels fast in our times, but not that folks tell me. But already discussed the why there.

The math is there. People just have to go look for it. And mathematics IS an infinite subject.

For those who love math? That is a reason for so much joy. You will never exhaust possibility.

And a reality check too I think, as how can people lead the future of mathematics in our world, if they can't simply discover?

Discovery defines mathematics. It is the base from which all else must build.

James Harris

## Tuesday, May 09, 2017

### Modular inverse innovation

Figured out a way to calculate the modular inverse of some residue r modulo some integer N, by leveraging a modular factorization. Will give the working system, show an example then give the derivation.

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

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

Example: Calculate the modular inverse of 11 modulo 137.

With N = 137, I will use m = 12, as: 7 = 144 mod 137

And looking at simple will use y0 = 1, so:

F0 = 11(11+2(12)(1)) = 11(35) = 111 mod 137

61d = 49n - 84 = 7(7n - 12) mod 137

If we now let d = 7d', get to 61d' = 7n - 12 mod 137, and n = -7 works with d' = -1.

So d = - 7, with n = -7, giving r-1 = (-7-1)(11+2(12)(1)) - 2(12)(-7) mod 137

So:  r-1 = -6 + 31 = 25 mod 137, and 11(25) = 1 mod 137 as required.

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

Derivation:

Given, some residue r modulo N whose modular inverse is to be determined, find D such that: m2 = D mod N, and m does not equal r.

Then for some  x2 - Dy2 = F, I have the modular factorization modulo N:

(x-my)(x+my) = x2 - Dy2 = F mod N, and can set:

x - my = r mod N, then x+my =Fr-1 mod N.

Solving for x, with x = my + r mod N, and substituting into x2 - Dy2 = F gets me:

r(r+2my) = F mod N

So I can look at a y0 which will give an F0, and a difference of d, such that:

y = y0 + d

And subtract that F from a multiple n, of my initial F, substituting for y:

nF0 - F = r[(n-1)(r+2my0) - 2md] mod N

Now consider, nF0 - F = 1 mod N, which gives:

r[(n-1)(r+2my0) - 2md] = 1 mod N

And can solve from there for the modular inverse:

r-1 = (n-1)(r + 2my0) - 2md mod N

Now can substitute into x+my =Fr-1 mod N, using x = my + r mod N, and simplify somewhat to get the control equation:

2mdF0 = [F0(n-1) - 1](r + 2my0) mod N

and from before I have to show the full system as given:

F0 = r(r+2my0) mod N

Derivation complete.

James Harris
-------------------------------

Want a PDF? You can get one at my math group. There is a post there with a PDF attached.

## Friday, May 05, 2017

### Chasing the modular inverse

For years I've talked about the easy modular factorization with x2 - Dy2 = F.

Which is, when m2 = D mod C, where C is some integer:

(x-my)(x+my) = x2 - Dy2 = F mod C

And then I'd use some residue r.

Where one way is: x - my = r mod C, then x+my = Fr-1 mod C

Where for years have wondered about that modular inverse, and found myself pondering, can you use these equations to figure it out? So why now? I don't know. Just playing around.

Next thing I knew was looking at C = 111 = 3(37) where picked for easy. And noticed that a close square is 121, so got m = 11, and D = 10.

Then went for a prime for r, kind out of habit, and need one coprime to 111, wasn't going to use 2, so went with r = 5.

So had: x - 11(1) = 5 mod 111, so x = 16 mod 111

Then I calculated F, from x2 - 10y2 = F, and got F = 24 mod 111.

So had 16 + 11(1) = 24r-1 mod 111, so 27 = 24r-1 mod 111

Which didn't seem to get me anywhere. But I noted means: 5(27) = 24 mod 111

So I'm like, ok, why not use y = 2? And after same process got: 5(49) = 23 mod 111

And, I realized could subtract one from the other to get 5(27 - 49) = 5(-22) mod 111, so:

5(89) = 1 mod 111

So I got the modular inverse, but was that luck? And now finally decided to go more formal, so here is the full system:

x2 - Dy2 = F, m2 = D mod C, where C is some integer.

(x-my)(x+my) = x2 - Dy2 = F mod C,  x - my = r mod C, then x+my =Fr-1 mod C

Solving for x, with x = my + r mod C, and substituting into x2 - Dy2 = F gets me:

r(r+2my) = F mod C

So I can look at a y0 which will give an F0, and a difference of d, such that:

y = y0 + d

And played around enough to know I needed a variable n, where will just give what I got without explaining everything in detail. Is easy math though, so not hard to retrace.

nF0 - F = r[(n-1)(r+2my0) - 2md] mod C

And chasing after modular inverse, nF0 - F = 1 mod C, would be nice, so have:

r[(n-1)(r+2my0) - 2md] = 1 mod C

And then:

r-1 = (n-1)(r + 2my0) - 2md mod C

Finally now then I have control equations with:

2mdF0 = [F0(n-1) - 1](r + 2my0) mod C

and

F0 = r(r+2my0) mod C

So now I don't even see the original x2 - Dy2 = F or the other equations. Just see those three, but originals must be valid for it all to work! Must keep that in mind. First was thinking oh, they just went away. Nope. Math knows they are there.

Well now can see how n = 1, and d = 1 worked, with r = 5 mod 111.

So was just luck, huh? So you CAN just say, pick n, but then have potential of needing a modular inverse to calculate d, or vice versa. Oh yeah, m cannot equal r for this approach to work! Should note that as well.

Also may as well check my new equation since know n =1, works, and clips off the front too.

And: r-1 = -2(11) = 89 mod 111, as we got above.

Went ahead and tried r = 41 mod 111, just to use something bigger, so again m = 11 and y0 = 1. And got:

105d = 3n - 66 mod 111, where d = 1 will work with: n = 57

And that gives r-1 = 65 mod 111, which works: 41(65) = 1 mod 111

And just playing around. So yes, can calculate a modular inverse this way, potentially. Definitely can get to another modular inverse to solve for n or d, but may not be necessary if can just figure them out like with my simple experiments. But who knows how often is that easy? I don't.

Wonder if could be proven.... Possibly something for some other time. These simple experiments have been rewarding and extremely interesting.

Glad did some experiments and some derivation.

Can be SO much fun to play with the math.

James Harris

## Saturday, April 29, 2017

### Thoughts on simplicity rules

Underlying reality itself, in SO many ways mathematics rules the world.

Theoretically as a human discipline mathematics should be the most objective, straightforward WITHOUT politics or social influence because mathematical logic does not care. But functionally have concluded that difficulty in understanding mathematical topics can bring those things BACK into the discipline.

But difficulty should only be there out of necessity, and not for social purpose.

So will admit would be so cool to flush mathematical industry of unnecessary complexity. And we should cheer simplicity and push the discipline away from worship of complexity as if it's a power to be not understood by most.

For me simplicity improves access and am sure helps reach. Am not someone who has the assumption that people will pay attention for any other reason than my math discoveries WORK.

Useless complexity though am sure can give room for charlatans to hide within mathematical field! And any such should be cleared to speed up mathematical discovery and progress. Humanity deserves that on the constant. And am sure science will need more advanced mathematical tools on the constant as well, where direction for tools with modular as a concept has unique appeal for its efficiency, simplicity and conciseness. Luckily have done lots of the groundwork over last decade plus which has a modular focus.

In number theory have been fascinated with realization that much of my own research relies on what I decided to call abstract reductionism, and how that really is about simpler approaches, including modular but not only.

For example one of my favorite results is the fastest way to count prime numbers for its size and complexity.

For those who wish to code, with positive ints or longs--where pj is the jth prime:

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

The P(x,n) function will count primes up to and including x, if n equals the count of primes up to sqrt(x), and if as you iterate you never let n be greater than the count of primes up to and including sqrt(x), if the function receives an n greater than that value it just needs to reset it to that count.

That is the fastest algorithm for counting primes for its size.

Turns out I noticed a neat trick which lets you use old ideas with an innovation that makes for a concise algorithm, but also leads to a partial differential equation and an explanation that eluded mathematicians for over a hundred years about the prime distribution. Simplicity ruled.

And abstract reductionism is even more powerful with what I think may be the first true modular algebra, which I introduced with a tautological statement, also known as an identity:

x+y+vz = 0(mod x+y+vz)

I call that a tautological space and with that I found a way to generally reduce binary quadratic Diophantine equations, where suggest you use web search to find if don't know where is on this blog already.

Finding by web search versus me just giving a link may be more impressive AND informative as can then see competing websites. And my way improves upon techniques put forward by Gauss himself! And as he is a HUGE hero for me that's a massive big deal. Simplicity rules again.

These simplifications though to me help make number theory more fun--and accessible.

Consider the irony, if some long, difficult mathematical argument which only a few people in the world understood--is simply wrong. And I dramatically demonstrated how it could even look correct under established rules of rigor.

Have concerns about to what extent hero worship took over the math field. People chase the appearance of correctness, daring someone to catch them if they're wrong, but oh, how do you know? In a complex argument which few understand how dare you challenge?

Shredding the incentives for empty celebrity I think helps mathematicians care for truth.

There may be simple ways to remove such.

Simplicity should rule.

That shatters political maneuverings, dismisses hero worship as basis for belief, and ends worship of position in mathematical fields, as who cares who figures out some math thing? I know the math does not.

People can, and it can lead them astray.

We should care in SOME way though I think. But what way is appropriate? Main thing is: irrelevant to mathematical truth who digs up that truth for the world.

Simplicity rules.

And simplicity has other benefits too, as being able to readily play with actual numbers is so great. Like my methods lead to a thing I wonder about usefulness. Consider:

42 + 6+ 10+ 14862 = 882

Which I copied from  a post where I demonstrate a general way to sum whatever number of squares you wish to get a square.

What use is it? Um, for me is just a way...um is fun. I guess. I've debated if should make more examples with it, but will admit...so maybe not so much fun for me. Maybe someone, somewhere out there at some time in human history will find it useful.

Main thing in my opinion is that you can just get to math. And will admit do cherish that I escaped the label of mathematician. The math does not care. Discovery should matter more than a label.

But I think it DOES matter when people can easily check a person. Overly complex and abstruse mathematical discourse in my opinion can often be about making that difficult, to force relying on people who know what they're doing--even when it may not actually be mathematically correct.

It can seem expedient to make a career over advancing your species.

I have no career as a mathematician to advance as am NOT a mathematician. And for me simplicity helps me do what I like to do.

And people should cherish valid knowledge. When you start looking at the person who discovered some mathematical truth, as if that decides if something is true or not--then you have lost mathematics.

Mathematics does not care.

Ultimately simplicity is a basic principle for obvious reasons.

James Harris

## Wednesday, March 22, 2017

### Valuing discovery and quadratic residue pairs

Years ago discovered there were these powerful control equations that force things like distribution of quadratic residues. Learning that happened at some point where was long enough ago not sure when, but for sure one of the important revelations was found fortuitously from staring at:

(n - m)2 = m2 + 1 mod D-1

which follows from:

(n-m)2 - Dm2 = 1

Should note don't use ≡ because I use modular algebra SO much and why add some extra line under equals all over the place? Seems like useless work to me. Oh yeah, to try and help those who might need the info wrote a post explaining mod years ago.

So was pondering a slight variation from the usual x2 - Dy2 = 1. And had been playing around with the different form and became fascinated by the forced quadratic residue there.

Very glad I noticed it!

Which lead me to finding out about quadratic residue pairs. Which I'd never heard of, until started searching to see if I'd found something new. So there, yes existence had been noticed before. Ok. Is cool to learn more historical number theory to research your own discovery. But then also I figured out how to prove some things, which was new.

You can kind of see things from usual form as well:

x2 = y2 + 1 mod D-1

And you can easily solve for unknowns modulo D-1 with either form.

Using my alternate: 2m = n-1(n2 - 1) mod D-1

Which you need for a proof, which is how to derive the count of quadratic residue pairs which for a long time was one of the most popular posts on this blog. And yeah can use it that way as IS one of the most powerful influences across integers, and yeah can use it to count. It is one of the infinity tools of number theory, which helps rule over quadratic residues.

Prior explanations for the count I found, using web search, relied on the pigeonhole principle. They had no clue.

For people into number theory, you can just go check! Web search. And is how I found much so I know information in this area is readily available. That reference reality is a major plus of our times.

So far as I know I gave the first direct derivation which not only lets you know why, it allows you to do more. Will admit am more into the 'why' as exciting, so just talked more you can do in my post as can expand to counting in more situations! But didn't do myself. But is a good example to see how a derivation can give more.

Especially is that odd feeling when you notice also an easy path that somehow escaped others. Why not figured out a century ago? Who knows. But probably because Gauss pioneered modular in his time, and I picked up where he left off, in ours.

Is odd too. Looks like in past, mathematicians knew more of modular arithmetic, but didn't realize a full modular algebra, which I did. And for me to get there had to create an entire mathematical discipline and pioneer abstract reductionism so maybe is harder than I wish to accept!

For me? Is just what I did. Guess that doesn't give me the perspective of looking at from a distance.

The joy of discovery I guess can be hard to describe but it definitely can motivate to find more, which I think is great. There is that thrill, for what YOU find, with your mind, and mental effort. Is a thrill...hard to describe.

Here again my modular approach leads to a short, powerful answer and in this case, a first derivation.

And it dawns on me that modular as a first reach in problem solving may be something very new to the human species, but why?

James Harris

## Tuesday, March 21, 2017

### Supporting faith in the system

Role I want to play can seem mysterious or even a bit complicated until you consider what I see as strange things. And to me the strangest things have driven others that seem so dramatic, like why I put forward a definition of mathematical proof, which I noted at the time back in 2005, was motivated partly by behavior from mathematicians.

And it's in the post what I found troubling, where had routinely seen in the news mathematical arguments presented as possibly proofs, which were called proofs, and mathematicians talking as if mathematical proofs were delicate. Huh?

Such a bizarre contention was about trying to explain to press, when things called proofs were later shown to have error, and I say, should call mathematical arguments being checked by the system to see if are proofs instead of calling proofs to the public before that is even done.

That behavior upset me. And surprised me too! Why talk as if mathematical proof was fragile or delicate? Why? Why? Why?

Do try to feel empathy for academic mathematicians desperate to make a career or just keep one going, but there have to be limits on selfish behavior, and constant focus on what is good for the system as a whole.

Political behavior from mathematicians or even worse, chasing fame with a claim is not an excuse for such disruption.

It irritated me. So I wrote a definition of mathematical proof, which has a functional basis.

Some of these things are SO wild, but true. It's like thinking about it later though, seems extraordinary. The explanations are worth nothing I guess. Usually I DO try to explain.

And, also note that gave me the benefit of checking my own mathematical results with it. Which I also needed as why wouldn't I?

With my one published paper, also note I've championed the system, as it passed anonymous peer review, and actually faced two reviewers.

Questioning their own system by mathematicians is probably why things stand as they are. If they just followed the rules would be so much easier for them. And I'm NOT a mathematician so I can critique that while wryly noting my detachment from that system is a plus.

Will also note my disdain for what have seen as quests for celebrity from certain mathematicians, and a disappointing amount of hero worship, which muddles the quest for truth from the established community. It's like math people want to believe based on SOURCE instead of argument. Naive hero worship worries me.

Yes, I have my own ways of making fun of such things, as well as highlighting them. I'm functionally focused on what works, as I test, and see over time. And is a process where I try to lighten things up when I can, to the extent I can. Helps me feel better at least.

I see myself currently as the biggest defender of the system, which is a responsibility I take very seriously. And that becomes more clear in time.

If you thought something else, why?

There is a LOT of hard evidence that I'm working hard to maintain certain things. Over and over again have gone to a lot of effort to support mathematics globally. Including taking the time to be careful as to how that is done.

There is no rush for me. Mathematics is important enough.

Feel like mostly done though, which is great! Heavy lifting? Over. Helping with mathematics is already mostly fond memories. Don't spend much time on these things now anyway. Just lately have been cleaning up a bit. Have other things to do too. Maybe have to do a little something else here or there if field of mathematics gets endangered again, but plan is soon NOTHING else, to fix. Can still talk things though. Is fun.

Often have been forced to face things objectively, accept facts, and realize I'm lucky to have a role to play, and not think too hard about how big it might be or how strange it is. As then can feel all kinds of weird, which is not fun.

Oh yeah, thankfully I haven't seen mathematicians trotting out mathematical arguments still under review as mathematical proofs that are delicate recently. If still happening, luckily not that I've noticed.

I find that behavior undermines public confidence in mathematics. And be certain I consider that to be unacceptable.

Mathematical proofs are NOT delicate. Never think otherwise.

Mathematical proofs are absolute in a sense few things human beings can ever find can ever be.

That will never change.

James Harris

## Friday, March 17, 2017

### Critiquing math people reality

Some tasks can be really difficult but really worth doing, even if you're not sure how it will all work out. Have learned to appreciate perspective and just working at things until done.

So yeah back in 2003 I wrote a paper which passed through anonymous peer review and was published, where luckily paper is still available hosted by EMIS as part of its archive. And that paper looks correct. No mistakes under established mathematical rules. But its conclusion contradicts with very well established number theory! So you have to use a separate argument to know the paper must have something wrong.

How is that possible?

Forget that, how can that NOT freak out mathematicians?

Since then I've noted, was demonstrating an esoteric and intriguing flaw with use of the ring of algebraic integers where that flaw is EASY to explain, but let's get back to that paper which has no errors under existing rules which can be shown to contradict with accepted number theory--how can that not just flabbergast mathematicians?

Some may know that there WERE hostiles to my paper, which is SUCH an ironic thing to me, as they attacked it for doing what it was supposed to do. As they pointed out that by a separate argument it had an incorrect conclusion. The paper itself appears flawless--unless you know conclusion must be wrong.

But um how is that possible? Can I ask that enough?

The very ability of a paper to do such a thing challenges so much. And such a paper actually does stand alone. It should not be possible.

The problem gives the potential of writing something that looks like it is correct, and checked against the established rules passes, which is nonetheless WRONG.

To me more than a decade since that paper can find a way to get a bit of dark humor in the situation, which is very serious.

Yes I critiqued the mathematical community--globally.

If YOU could, would you?

Did you pass? Or fail?

Why would I test mathematicians all over the planet in this way? Why wouldn't I?

It was a unique opportunity. Such an opportunity may never exist on this scale again in human history. Test mathematicians all over planet Earth with something THIS simple? Too cool.

Yes, have struggled with the weight of it though, at times. Mathematics is so important for our world. Wondered the fate of our world if I failed. Wondered if I could fail--if reality would let me. Such concerns seem distant now.

The test is perfect. Proper reaction is to understand that until fixed doubt can exist throughout the field of mathematics, though the problem does not exist for arguments that rely on fields. It only exists for number theory relying on the ring of algebraic integers.

And it IS checkable by any mathematician as well as uses elementary methods. So a person doesn't have to be a number theorist, and other mathematicians have no excuse. And yeah I'm NOT a mathematician I note again. Is important, you know? Helps my detachment.

Notice that the test remains until fixed as any math student can simply check the paper, and read through noting--no error with argument under existing rules. And then simply note that conclusion is false under existing number theory.

Until things are fixed of course and is simply a matter of historical record and example is part of the curriculum for future math students.

I have given the fix of course, which lead me to putting forward the object ring.

So yeah the conclusion is wrong in the ring of algebraic integers but IS correct in the object ring.

Oh, so how easy to explain? Imagine someone says they are only using evens and considering factors, so yeah 2, 4 and 8 are ok, and factor easily enough. But they trot out 6, and you say, NO! Because now with those rules 6 and 2 do not share factors because 3 is odd. And yeah evens are NOT a ring. And the ring of integers does NOT have this problem.

But the ring of algebraic integers DOES have an equivalent problem which I've shown multiple ways and even given a full explanation lately. as apparently I just LOVE explaining it. So the problem with the paper is the ring declaration with which it begins.

So why has the global mathematical community failed my simple test on the whole for so long?

I do have theories.

But not everyone really failed. My paper passed two anonymous reviewers and was published. Was chief editor who tried to pull later. And EMIS has kept it up. Am sure there are plenty of people out there who...I don't know. Maybe just figure there should be.

And it could still take awhile. What's good now is EXPLANATION. For those who have wondered you can now finally see the complete picture that took about 13 years to fully get outlined by me. So much work too! But learned so much has been worth it.

Now you can read not only the point of that paper that got published but see a full mathematical system explaining it all out in extreme detail. I think explanation is awesome. At least now people can know why.

Reading through seems so dramatic though. Yup. And we are in the 21st century with new ways.

To me am in a functional process where I see what works, and also it helps me to handle it all.

Figuring it all out is a challenge so worth the effort.

After all, our world's knowledge is what is important. And mathematics is important enough.

James Harris