Saturday, December 27, 2014

Celebrating simple math

People naturally have preferences, and preferences can shift what one sees--or can see. Then certain things can simply fall between the cracks. Which is why I'm proud to celebrate simple math. Past mathematicians had the thrill of these types of results, but so can we.

And that pleasure can be had by anyone including those NOT mathematicians, as I'm not one.

Maybe that's why I can find things that well-trained modern mathematicians I think have been indoctrinated to think do not exist. There's some notion that all the good simple discoveries were already discovered long ago.

Like recently I had the pleasure of offering a challenge based on a simple solution, giving those who like to play with numbers a chance to learn something not known to greats like Euler and Gauss, or was it?

The other thing you get from a challenge presumably is people digging into the archives, and I'm curious! I want to know. Will not hurt my feelings if Gauss and Euler knew of it. As I still re-discovered in that case meaning I get that precious thrill that few ever get to experience.

And discovering an infinity result with THIS level of simplicity offers that rare thrill that few human beings will ever get in the history of our species.

And I'm proud to dedicate this result to my parents, even though they think this math stuff is nuts. And once it's verified to be unique to me--if it is--to my country. My story is so much, only in America. But that's getting ahead of things. Someone may have had this thing before. Will take time to see if it's an original result.

The other thing you get with an excitement over just playing with numbers is that enthusiastic need to show over and over again, so here's another simple list, which follows from the general result.

Let's let m = 17, and u = v = 1. Then, the BQD iterator is:

(u - 16v)2 + 16(u + v)2 = 17*F

Start is:

12 + 16*12 = 17

then it must also be true that

(-15)2 + 16(2)2 = 289 = 172

Next iteration: (-47)2 + 16(-13)2 = 4913 = 173

And third iteration: (161)2 + 16(-60)2 = 83521 = 174

Fourth iteration: (1121)2 + 16(101)2 = 1419857 = 175

Fifth iteration: (-495)2 + 16(1222)2 = 24137569 = 176

Sixth iteration: (-20047)2 + 16(727)2 = 410338673 =  177

---------------------------------------------------

And in general: x2 + 16y2 = 17n+1

Or you could just say: x2 + y2 = 17n

If you were going for the ultimate in pretty simplicity. Where x and y will always exist, where they are nonzero integers, where we know y will always be even, and n is a positive integer.

Interesting to look at this thing with somewhat bigger numbers. I do wonder if Gauss or Euler or anyone else for that matter besides myself figured this thing out before me. My suspicion is evidence strongly suggests they didn't, as I've used this result to explain and extend a result from Euler and from Ramanujan. It also gives a new way to analyze Mersenne numbers. If they had it, they'd have used it like I just did. Broad reach is often an indicator of how fundamental a result is.

And not claiming it's complicated of course. Celebrating simple here! But it may give a reminder of just how HUGE mathematics is. Mathematics is an infinite subject. We get lucky with pieces of that infinity.

Of course being trained not to find simple results may affect a math student, but has no meaning to someone like me, as I'm NOT a mathematician. I got physics training. I was actually surprised to read that mathematicians believed all simple fundamental results had already been found. I never received that training.

So the future of finding these types of results may be about non-mathematicians. Which I think is ok.

Maybe that's actually better, at least for someone like me! As if certain attitudes hadn't taken over the mathematical community, the result may not have been left around for me to find.

Want to help support basic math research of this kind? Then please talk about it. That's the best thing.

Have also a discussion group which should mention: My math group

James Harris