Friday, October 28, 2016

Why focus on two conics

Was fascinated to realize that an ancient expression gives quite a few conic sections as it in fact gives 2 of 3:

x2 - Dy2 = 1

Where depending on the sign of D, you can get an ellipse, which includes circle for D=-1, or a hyperbola for positive D. So I like to call it the two conics equation.

But have noted it as giving 3 conics as some consider the circle to be one in its own right, but circles are ellipses. And am sure is something I considered years ago before focusing as I do.

And that's with fields of course. The equation got lumped into the category of Diophantine where the hyperbolas got all the attention, though only looking at integer solutions, which is a historical oddity I think which traces back to Fermat as to how that emphasis stuck. The equation has been known for thousands of years though.

For a bit got excited as wondered if I should really go with three conics but I thought over so many things carefully years ago. And have learned the hard way to trust myself. There are times it takes me days to remember why things are a certain way. Luckily here was a matter of minutes.

James Harris

Sunday, October 23, 2016

Naming things is hard

One of the best things to me about what I've learned is that I can unequivocally say that people don't need to mess with what is commonly called Fermat's Last Theorem, as Gauss was right about it. And I think in mathematical history it will be seen as an oddity more important for the effort around it, as the result itself is trivial for the reasons he gave.

And for me it is relevant because it was SO frustrating that I invented my own mathematical discipline to tackle it which I eventually called modular algebra symbology.

For a while was still looking at it with that then was like, oh my.

And I realized I needed the object ring.

And modular algebra symbology explores what I decided to call tautological spaces, where you do analysis on what I decided to call the conditional residue.

And with modular algebra symbology I came across something I decided to call the binary quadratic Diophantine iterator or BQD Iterator for short.

Also years ago decided I had found an axiom previously not accepted to be one! And decided to call it the prime residue axiom.

Oh yeah and renamed something the two conics equation as can give hyperbolas or ellipses, and its generally used name, at this writing, is considered to be a historical error.

And remembered that I found what I like to call a quadratic residue engine which is key to derivation of expressions to count quadratic residue pairs.

I think that's it. Not interested in continuing to update as have a couple of times, and still maybe have named a few other mathematical things here or there but I think I got my most important namings.

James Harris

Saturday, October 22, 2016

How much a global resource

So far this year according to Google Analytics the blog has had visits from 298 cities in 50 countries, by people with 36 languages, so suspect Google Translate does help.

Those numbers show a global interest reality which is greatly appreciated.

Oh yeah, guess may as well give last year for reference.

Last year it shows visits from 423 cities in 68 countries and 43 languages.

To get these numbers went into Google Analytics and set the dates, and then clicked options to get information and read off what it showed.

There is something about mathematics.

Thanks for the interest.

James Harris

Tuesday, October 18, 2016

How do I know?

The appeal of human authority is great and for most people that's all they have. They are told what is true or not, and in many cases have no means to check. Human authority is the ultimate authority for them, but in mathematics that is not the case.

But MOST of what most people know of mathematics IS on human authority while I found myself metaphorically in the wilderness with new mathematical approaches which was VERY distressing. So I found absolute truth in identities.

The identity which changed so much for me I call a tautological space:

x+y+vz = x+y+vz

Is weird to talk how much I considered that, and thought about it, to convince myself that what I was taught by human authority and common sense must be true, actually was! Identities are true in and of themselves, like x = x. Is just true. But I was doing different things with it.

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

Wrapped up into a modular expression is the SAME THING but presented differently. Is weird though. Some may think that 'mod' means the two expressions are different. Puzzle that one out then. (If it is new to you, I explain more on this page.) Why are the expressions equivalent?

From there it was just a matter of believing in logic, and over time I relied on mathematical authority, and human authority? Well that can just be plain wrong.

And I learned a love of mathematical proof.

James Harris

Tuesday, October 11, 2016

Why proof matters

Mathematics and logic are the human disciplines which can rely on absolute proof. And I think it interesting to consider the possibility of people who don't even believe that exists. But it is a cool thing which theoretically removes the problem of conflict with mathematical ideas. Or as I like to say, proofs don't fight.

However there can be things subject to interpretation, though functionally I went ahead and presented a method for always determining mathematical proof which can be logically determined to be perfect, which is to begin with a truth, and proceed by logical steps. At each logical step, you have a truth, so my functional idea was to connect truth to truth with logic connectors. Oh, you may wonder, but how do you know you began with a truth? Good question. Um, asking myself and noting.

And I made an example to demonstrate just so can point to it when discussing:
Example post demonstrating absolute proof

Which I think is handy.

The example shows that with the condition that x2 + y2 = z2, then

(v2 - 1)z2 - 2xy = 0(mod x+y+vz) must be true, where v is a free variable.

So, since v is free, let's say I let v=10, then I know that 99z2 - 2xy has x+y+10z as a factor, when x2 + y2 = z2.

Like if x = 3, y = 4, then z = 5, and 99(25) - 2(3)(4) = 2451, and 3 + 4 + 10(5) = 57, and 2451 divided by 57 equals 43.

You get trivial results like that here but can use this approach to probe into LOTS, where v is a tool of your mood. I like that, and in fact picked 'v' for victory, mostly.

That result is absolutely true where those expressions are available, without regard to ring. I like that.

Oh, and I made up that mathematical analysis path, and like to call the mathematical area--modular algebra symbology. For some reason now that naming...well is accurate. Naming things is fun but at times I wonder at my choices. Where use what I call tautological spaces, where advanced the concepts enough to cover all the mathematics interesting to me.

But yeah to me modular algebra symbology is SO cool, and it helps that I made that up, so is my own personal mathematical discipline, but others can use of course but for me will always feel different. And yeah I pioneered a functional approach to determining if you have mathematical proof, so it feels different using it too.

Web search is an AWESOME way to check a person on bold claims. Easy is to just search on modular algebra symbology, as yeah I did make that up. Also you can search on tautological spaces, as I made that up too! While checking me on how people check mathematical proofs is where the hardest work may be, but search on that too where will not suggest searches. If interested, figuring out how to search to check me on functionally defining mathematical proof is useful as well.

Because mathematics can allow absolute proof it does make it kind of boring to argue it, and I quit that years ago, though will admit that the utility of arguing about mathematics is about finding error. So it DOES have value, arguing about mathematics, but in my opinion, the only value is in finding error which means a mathematical argument is NOT proof.

Sure someone can CLAIM absolute proof, but does that same person actually have it?

Turns out I figured out how to check, and yup, do use it! Is fun.

And yes, to me saying absolute proof is redundant but is useful for emphasis and clarity.

Proof lets you move on and do other things, like find more mathematics! Or hang out. Do something entertaining, or some other kind of work. Or, whatever.

James Harris

Saturday, October 01, 2016

When derivation is surprisingly easy

One of my favorite and more popular results is a derivation of the already known way to count things called quadratic residue pairs, but with a derivation can go beyond what was known before it.

For those curious about what a quadratic residue pair looks like, here's a list of quadratic residues for 31:

1, 2, 4, 5, 7, 8, 9, 10, 14, 16, 18, 19, 20, 25, 28

And there are 7 pairs: {1,2}, {4,5}, {7,8}, {8,9}, {9,10}, {18,19}, {19, 20}

And that count is given by: floor((31 - 1)/4) = 7

Deriving that is easy, and fun!

Turns out MAYBE though I took a slightly long route as you can look at:

x2 - Dy2 = 1

Which I like to call a two conics equation as it gives hyperbolas or ellipses depending on the sign of D, and look at it modulo D-1:

x2 - y2 = 1 mod D-1, so:

x2 = y2 + 1 mod D-1

And you can notice easily enough that a quadratic pair can be there though the full set of rules for when it is, are not obvious there. Like for D-1 = 12, there are none. So how does the math do that? I think y always has prime factors of 12 or something, and um, it isn't very simple. Check out my post linked above for the derivation where I explain a LOT, as just sitting here now am like, it's not THAT simple.

If you can, you might want to try yourself! See if you can derive a quadratic residue pair count from here, or even more fun, find a derivation out there in the wild. That's kind of a trick thing to put though as I found the first, and to my knowledge only derivation. Do some research to see how the result was found before which if I remember correctly involved something called the pigeonhole principle.

Oh yeah, so I used "mod" so much I quit using a congruence symbol years ago, and don't think I lost anything. Also for those who'd like a primer for me on the subject, wrote one in a blog post years ago:

Focus on modular arithmetic

I really think modular is one of the coolest things ever, and of course can show up beyond mathematics. In mathematics though modular concepts lead to astonishing simplifications. I like to say, modular algebra gives a handle on infinity.

James Harris