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Tuesday, August 11, 2015

Infinity and binary quadratic Diophantine equations

Possibility is amazing. How do we know what we don't know? But we can have clues from what things people find that are very surprising. Like one of my discoveries years ago was proof that binary quadratic Diophantine equations are in general connected to an infinity of binary quadratic Diophantine equations where my discovery relies on using an identity.

So I found years ago a way to reduce binary quadratic Diophantine equations, which in general look like:

c1x2 + c2xy + c3y2 = c4 + c5x + c6y

You could reduce to a simpler form, where with my method you do NOT care about the discriminant. Turns out the discriminant is unnecessary when reducing these equations, and 'discriminant' is a technical word familiar to people taught classical methods to reduce.

It being unnecessary to me is really cool! And is one reason my way is simply better as I've discussed recently, because I bring it up routinely.

To put things into some perspective, number theory students who have learned about the discriminant for reducing, can imagine a future where students are not even taught it, in that context, as it's unnecessary. Which it is.

I've tried to usually link to one post for my method to generally reduce, where I use a variable 's', which is a function of x and y, where it was ego that I don't give it there.

Turns out that: s = (c2 - 2c1)x - (c2 - 2c3)y - (c6 - c5)

Which was determined by math software as it's just too freaking hard to figure it out by hand, which I started trying to do a couple of times and never finished, so I wouldn't give the value. Others used the math software I should add. I've decided I was just being silly.

The reduction method gives you a reduced form, but if you use it against a reduced form, it gives back a reduced form, as you can reduce no further, which looks like:

u2 + Dv2 = F

when reduced you get

(u-Dv)2 + D(u+v)2 = F(D+1)

Which I've had for years but recently decided maybe I could jazz up interest by naming it, so call it now a Binary Quadratic Diophantine iterator, or BQD Iterator for short.

Oh yeah, so now we've connected almost all binary quadratic Diophantine equations to an infinity of other ones. Turns out there are some exceptions, but they are trivial cases.

Check out one of my fun things that I did with the result years before I named it:

Let F = 1, u=x, v=y, and D = -2.

1. x2 - 2y2 = 1

2. (x+2y)2 - 2(x+y)2 = -1

3. (3x+4y)2 - 2(2x + 3y)2 = 1

4. (7x + 10y)2 - 2(5x + 7y)2 = -1

5. (17x + 24y)2 - 2(12x + 17y)2 = 1

and you can keep going out to infinity, but I'll stop with 4 iterations.

Notice now you can use the simple case of x=1 and y = 0:

1. 12 = 1

2. (1)2 - 2(1)2 = -1

3. (3)2 - 2(2)2 = 1

4. (7)2 - 2(5)2 = -1

5. (17)2 - 2(12)2 = 1

And you have answers to x2 - 2y2 = 1 and x2 - 2y2 = -1.

Which I like to trot out. To me it's just AWESOME.

I really don't know if that series is in an established number theory text, but it does use my BQD Iterator, which appears to be mine alone, so I wonder how it might. Of course someone could have got lucky, just playing around. But then they'd have no clue why that worked!

So yeah, my way of reducing binary quadratic Diophantine equations just throws away the discriminant. And works on any such which is not trivial. That is, you can find ones where my method doesn't work, but those are easily solvable, and equivalent to a unary case anyway. Like the example I usually give:

x2 + 2xy + y2 = c4 + x + y

is: (x+y)2 - (x+y) - c4 = 0

Having the BEST way to reduce these things is guaranteed attention. Turns out they have practical uses, so there is no doubt that people will find this method and use it.

And since the method was discovered with an identity there is no doubt about correctness.

It is perfect. Absolutely perfect.

That an identity can be used in this way is fascinating to me. Finding the identity requires more advanced techniques which to me is just more modular algebra symbology.

If I had done nothing else in my life just this discovery alone would give me my place in mathematical history. So it's like, I'm good. And of course got LOTS of other things, many of which are much, MUCH bigger. So I'm like, chill.

If you believe in mathematics then it's a simple situation.

So I found something cool, but who knows what someone else might find? We don't know what we don't know, you know?

Mathematics is an infinite subject. To me it is a no-brainer that it has an endless ability to surprise us, keeping the spirit of adventure alive. We can never fully encompass the infinity that is Mathematics.

The web links to my pages but years ago I changed the name of this blog, and the web promptly linked to pages on Usenet where I'd discussed it. Until the blog regained position in web search under its new name of Some Math, and the web links switched back!

Our world is hungry for new useful knowledge. It will do what it takes to have and keep it.

That fascinates me so discussed it more than once, where here is a highly analytical post.

But even without them I'm sure plenty of people all over the world have copied the method down, so it's a permanent part of the world's body of knowledge.

And with it comes the stunning reality that in general binary quadratic Diophantine equations are connected to an infinity of other ones, like some massively huge freaking number theoretic family.

You can't stop the connection. It rules.


James Harris
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