Translate

Thursday, December 07, 2017

Explanation and unary two conics equation

Much to my surprise came across a simple explanation for size of integer solutions to:

x2 - Dy2 = 1

I like to call it the unary two conics equation, as depending on sign of D, you can have ellipses or hyperbolas. And traditionally people in mathematics like to attribute it to a guy they also say should not get credit, which is SO wacky to me. Why not just correct the error? I like name I give it better. And traditionally its integer solutions are of interest as solving it rationally was known to Fermat, who liked to play intellectual games looking for integer solutions.

And integer solutions can seem kind of random, if you don't know the why, like consider:

98012 - 29(1820)2 = 1 with D = 29

But next one:

112 - 30(2)2 = 1 with D  = 30

Giving smallest positive non-trivial integer solutions.

And turns out that key to size is D-1, and not D unless is prime. Which is the kind of mathematical wrinkle that can confuse humans I guess? Which is why wasn't just noticed centuries ago? But even if they noticed, how would they explain? I can explain why.

Also people have to be curious. I've tried to research this subject and apparently number theorists gave up on finding an answer long ago so claimed there was NOT one. So no way they'd ever find one, eh? So primitive to me, how some people choose to think a certain thing, in order to not learn, satisfied with a belief. So was declared no answer existed. Interesting.

Where lots of the easier solutions are going to be when D-1 is prime. And hard starts if has small prime factors, and harder the more of them you have. Like if D-1 has 4 as a factor and is not square? You start getting the bigger ones like example I showed, where also is key that D is prime.

There are also other things make for easiest answers, like trivially found solutions when D-1, D+1, D-2, or D+2 is a square, as well as when D is divisible by 4, for when D+4 or D-4 is a square, where kind of didn't even wish to give all that but figured why not for completeness. Oh yeah, so smaller integer solutions for those more boring cases as well.

The general result around factors of D-1 is vastly more interesting to me.

With explanation in hand just jumps out at you if you scan through tables of integer solutions. For me? Was so wacky wild years ago. But of course for me now is old news. And also, I discovered so my emotional relationship with the information is different.

The result then is that D-1 is the control in general for size of solutions, and its prime factors as to how small and how many are what matters. Turns out the math has to work harder for integer solutions if D-1 has lots of small prime factors, especially if D is prime. And hardest if D is prime, D-1 has lots of small prime factors and 4 is a factor, except for the exceptions with various squares, like if D-1 is square. Which then was the previously unknown explanation. Easy answer which is backed by mathematical proof!

Is slightly convoluted though when all the details are laid out. So yeah, D-1 is most important, but D impacts also if prime! And yet there are all these exceptions when you get easy, like D plus or minus 1 or 2, and yeah, I guess it could be hard to tell by looking over solutions trying to guess. And results could look random to a human without all the rules?

Where copied from this post used as reference. Have another post soon thereafter talking two conics equation size which is way detailed.

Is very interesting to me these relationships between integers which control things. There is that mathematical precision, of course, to the machinery of integers in relation to each other.


James Harris

No comments: