I'd been studying modern problem solving since I was a kid, and realized that classical math problems considered hard offered a unique opportunity, as I could work on them indefinitely with no reason to think I'd succeed.

That might seem strange to deliberately look for problems you don't expect to solve, but that guaranteed I could work on them indefinitely. Sometimes I've likened it to a dog gnawing on a bone--the point is the effort.

And I also knew I had to limit myself, so decided to focus on what mathematicians call elementary methods, while I emphasize I am NOT a mathematician. And I have no interest in becoming one either. I just wanted the kind of thrill I'd had through a lot of my life as a supposedly smart kid. And at 25 years old I knew I didn't want to do graduate studies, as I was done with school, and didn't want to get pulled into additional formal academic study in any field.

That was back sometime in April 1995 and nearly 20 years later I'm glad my younger self sent me down this path, but little did he know I'd someday have a result like this one:

**x**

^{2}+ y^{2}= m^{n}will always have nonzero integer solutions for x and y, as long as m - 1 is a square and n is greater than 0.

For example:

**495**

^{2}+ 4888^{2}= 17^{6}That result looks like something from antiquity. But I found it.

Though the search is still on to see if anyone else discovered it first. Which has its own benefits, as I solved a related problem because I was searching the web looking.

And it is not like my discovery is hard once found, as calculating x and y is easy, and you can see for yourself in a post where I talk about celebrating simple math, and muse about why the result was available to me to discover.

And I strongly suspect it is my discovery alone as I used the underlying mathematics to explain a result from Euler and a related result from Ramanujan, as well as extend from them, and they are HUGE in mathematics. There's almost no scenario where if this thing were already known that wouldn't have been done before.

Like, who wouldn't want to be able to say they not only could explain a result from Euler and Ramanujan but extend it as well beyond what those two great minds started?

And it's not the only thing I've found but presents well and even it is backed by over a decade of research.

Treasure Island, San Francisco, CA, USA April 20, 2010 |

Not being a mathematician probably helped me ironically enough. While being motivated to just grind at problems primarily as mental exercise did as well, and using elementary methods kept me from getting bogged down learning techniques versus inventing my own.

It's weird: sometimes constraints no matter how arbitrary can make all the difference as they guide you in a particular direction.

Resistant to using complex methods by my own choice, I was pushed to range more widely for simple ones. Not concerned really about success--though I do like it--I could keep at it for the sheer enjoyment, even in the face of 'haters' as I think motivation is very important. And some of the nastiest people out there simply have the wrong motivations.

Reality is I've had fun beyond my wildest expectations and kept my mind occupied, though lately this hobby is becoming a bigger deal as I consider what responsibilities I have to others.

I guess that's why some things are shifting as I approach 20 years of this particular intellectual pursuit and consider, now what?

And I'll readily admit I'm struggling now as I try to figure out what responsibility to others there is. Worse I'd actually like to succeed so it's not like I can just shrug off failure like I could with the math. Right now the social is mostly a mystery to me. I'm not sure what's going on, or why, or what I should do.

The math research doesn't care though, so I have no responsibility to it. A result like the one I show in this post quite simply does not need me.

A hundred thousand years from now if some being we can't imagine is reading these words that entity could appreciate my mathematical result, as it would work just as perfectly: pristine, absolute, incorruptible.

There is an odd beauty to these simple mathematical expressions and a sense of timelessness.

I think of mathematical results as like beautiful gems buried all over the place, as there is an infinity of them, and they do not care who discovers them any more than physical gems.

It's a great thing.

Playing around with simple math is a great way to exercise the mind.

I highly recommend it.

James Harris