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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, 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 which 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 the difference equations leads to a partial differential equation. That is so cool. And celebrate 15 years this month since first found.

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 humanity 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 my research shows 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 established 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 + 362 = 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.