x2 + (m-1)y2 = mn+1
Has non-zero integer solutions for x and y, for m = 3 or higher, and n = 0, or higher, and I have m raised to n+1 so that n is a count of iterations, if you want m raised to the nth power you just start n at 1 instead of starting at 0.
For example, with m = 5, and n = 6, x = 29 and y = 139 is a solution:
(29)2 + 4(139)2 = 78125 = 57
Of course the 4 can be pulled into the square so you have a sum of squares:
(29)2 + (278)2 = 78125 = 57
The necessity of answering reasonable concerns has helped me find the fun as well. As some people LOVE that "gotcha" thing where you find something you think is cool and they toss something back at you to try and make you look like an idiot or a sucker. Critics abound.
And I finally addressed one of the biggest insults tossed my way, which is the horror, horror, horror that I once worked on trying to solve Fermat's Last Theorem!!! And yeah, I'll admit I find it fascinating that certain math people feel so calm about that as an insult as if it is one of the ultimate terrible things that you can do! It's weird. And I haven't worked at it in over a decade, but that one seems to be so useful that people keep using it. So yeah, before you talk to anyone about my research you need to know that one, so when they throw it at you, you can say, oh yeah, know about that....
To me though the most important thing was sharing an example of a mathematical proof that demonstrates how you can know absolutely that it is correct. That to me is actually a lot more important than the social stuff, as it tells you how to be confident regardless of what people believe. As people can be supremely confident in completely wrong stuff.
Probably ended the year out strongest by the more spectacular, as giving an original basic research result that also can answer and expand upon Euler and Ramanujan is about as big as it gets. But back in March with my second post of the year I talked about my mainstream concerns. Which also explained the name change of this blog.
It may seem strange how often I make it clear I'm not a mathematician, but I think that is important. And each person can consider their own personal beliefs about what kind of person can make an important mathematical discovery. I suggest that the math doesn't care.
Our beliefs can be so important though, which is why I accepted the responsibility of sharing certainty this year. It's not enough for me to have all the fun. And it's not fair for me to leave others defenseless to people trying to mislead them about my work.
Sharing certainty really was the theme for the year, and that is also about sharing the joy of discovery, with defense against those people who will try to ruin it for you. And I think that's a good thing and a responsibility I should take seriously. This year I did.
So what about next year? Well, we'll see.