Thursday, August 18, 2016

Reducing a quadratic Diophantine to find solutions

Will readily admit do some posts to talk things where it also helps when I just get curious as well. As found myself thinking should do something new showing my method for reducing binary quadratic Diophantine equations, and eventually decided to go with:

x2 + y2 = xy + x+y + 102

There is a LOT of deliberate easy there and found out could still get more difficult with that last.

Where it took me a bit to pick that where I was too ambitious at first and tried 1000 which had NO integer solutions! Played around there for a bit and decided to use smaller numbers and 100 and 101 had no solutions. Luckily 102 did, as get tired of such things quickly.

(If you want to test your math knowledge, find solutions before reading further.

Can you prove that with 1000, 100 or 101 instead of 102, no integer solutions?)

Oh, and first thing is to standardize to the form I used with my method to reduce! As forgot to do that at first and was confusing myself:

x2 - xy + y2 = 102 + x+y

What follows relies on my post showing how to reduce and you can reference it to see how I get A, B and C as well as the rest.

Ok, so then A = 9 + 4(-3) = -3, and B = -3(0) + 2(-3) = -6, C =-4*102(-3) = 1224

Which means:

(-3(x+y) + 6)2 + 3s2 = 36 +3672 = 3708

And dividing off 9, gives:

(-(x+y) + 2)2 + s2/3 = 412

And -(x+y) + 2 = 20  works, as gives s = 6, as a solution. I like to use the positive solutions as just want an answer. This thing is picky though! Was maybe a bit surprised had to search a bit, but glad 102 wasn't too far from where I started.

x+ y = -18, so y = -x-18, and substituting with original:

x2 - x(-x-18) + (x+18)2 = 102 +x - x - 18


x2 + x2 + 18x + x2 + 36x +324  = 84

3x2 + 54x +240 = 0, which is: x2 + 18x +80 = 0

And (x+8)(x+10) = x2 + 18x + 80

So have two possible solutions and will use x = -10, so y = -8. Oh there's a symmetry thing going there.

So original was: x2 + y2 = xy + x+y + 102

And: 100 + 64 = 80 -18 + 102 = 164

And that's just one set. Noticed also that s = 33 works. And can get:

x = 3, y = -8

9 + 64 = -24 + 3 - 8 + 102 = 73

Is interesting to me that coming back to play with the math often for me is an adventure, especially as get some distance from the discovery.

Here it also kind of intrigues me that there are NO integer solutions for:

x2 + y2 = xy + x + y + 1000

Had a vague feeling that some combination would work, and that vague feeling was wrong. And then there weren't any integer solutions with 100 or 101 either.

Those integers can be SO picky.

Numbers just stay interesting to me.

James Harris

Friday, August 12, 2016

Simply interesting across centuries

Will admit one of my more gratifying finds can be shown with some really simple examples, where luckily noticed that both Ramanujan and Euler had shown interest in the area.

Here are some expressions:

12 + 7 = 23

Next : 32 + 7 = 24

And next is: 52 + 7 = 25

And then: 12 + 7(3)2 = 26

One more: 112 + 7 = 27

Notice that 7 is bare, in all but one case.

And according to MathWorld, Ramanujan has a result called Ramanujan's square equation:

x2 + 7 = 2n

Which they say only has solutions for n = 3, 4, 5, 7 and 15.

And I've shown the n = 3, 4, 5 and 7 cases, but am not interested in iterating 8 more times to get the last one.

And also on MathWorld I found that Euler had an interest in this area as well with his own solution which was NOT using my BQD Iterator. Will admit have no clue how he found it, as is something more general.

Where I explain more in a previous post where I note these as validating historical connections.

My methods generalize to allow you to solve a much larger variety, ok an infinite variety of similar. Will admit that being able to explain with something that Euler probably didn't know and that Ramanujan probably didn't know in an area where both showed an interest feels weird. Am doing better with it now as time has passed. You just kind of feel good after a bit and lose most of the overawed feeling.

That celebrity aspect does make me ponder a bit, but will admit is good fun.

James Harris

Thursday, August 11, 2016

Simplifying understanding works best

Years ago I learned the hard way just how important simplifying, simplifying, simplifying is when you're trying to figure out your own math. Repeatedly I'd had mathematical arguments I thought brilliant, until after making them simpler I'd watch fall apart with what then looked like silly errors.

So I learned to love simplicity, and also to dread it.

It is an odd thing when you look at something you found, maybe even years ago, and realize it has simple implications you have not checked.

Mathematics is brilliant in that everything has infinite consequences some kind of way.

Infinity creates the highest standards.

That would fascinate me at times, looking over mathematics I found knowing it has to work over infinity. And scare me if I dared check something I hadn't before. And then that odd feeling when the mathematics works perfectly.

There is nothing like it.

If you can simplify? Do it, if you dare.

James Harris

Wednesday, August 10, 2016

Covering the blog with interest

Feel like I do need to emphasize am NOT a mathematician, and there are some difficult aspects to this situation, where feel like have important mathematical ideas without establishment support. But so glad for the interest, and trying to be more appreciative of people who really have supported me with it.

So this blog which started March 20, 2005 has people STILL reading blog posts throughout that history. And some of the most popular posts according to Blogger are years old! That is amazing.

Though yeah most interest is concentrated more recently which is also good, as like to think have learned a few things over time, and more refined results are more recent as well. Before was spending MOST of my effort at basic research.

And will note have ended basic research as of July 31, 2016. Which I posted about on one of my other blogs.

Being a very curious person have been gratified to find blogs help me structure around my interests and keep up with things! And also of course blogs help you efficiently share with the world

Ending the basic research process lets me focus more on refining ideas, or just enjoying the ones I have.

Thanks for continuing interest. Yes, I do notice. Much appreciated.

James Harris