Tuesday, December 13, 2016

Summing squares to powers

Sometimes I just like to play with numbers:

12 + 9*12 = 10

82 + 9*22 = 102

262 + 9*62 = 103

282 + 9*322 = 104

Which goes out to infinity, where I used a simple rule.

If: u2 + 9v2 = 10a

Then: (u - 9v)2 + 9(u + v)2 = 10a+1

Which is just using my BQD Iterator. And notice that u and v can be positive or negative, while I like to show positive as is easier and looks prettier, which allows some selectivity, which I used behind the scenes to get my series above. Some choices made it more boring.

Like yeah so if you use a = 4, u = 28, and v = -32, then next is:

3162 + 9*42 = 105

So yeah, every power of 10 can be written as a sum of two squares.

And was fascinated a couple of years ago that the general result is, for an integer n equal to 0 or higher, and an integer m equal to 3 or higher:

x2 + (m-1)y2 = mn+1

I have m raised to n+1 so that n is a count of iterations. And if m-1 is a square then every power of m can be shown as the sum of two squares.

And talk it all out in this post. So I just used m = 10 above.

James Harris

Sunday, December 11, 2016

More thoughts on coverage

This post is labeled behind the scenes as maybe isn't so interesting for those looking for other things, like math. But is useful for me to record things in place I like to say.

Have realized some posts which talk the coverage of the blog in terms of registered visits could be really useful later. So with year closing out found myself in Google Analytics this morning running the numbers so far, focused on location.

And according to Google Analytics blog had visits from 57 countries so far this year, consistent with what posted earlier on this subject. Have a map of the world can look at and see whitespace where no visits, and as expected most of that was across continent of Africa. And looks like cover about half of South America.

The city counts kind of fascinate me now, and had visits according to Google Analytics from 346 cities, where can switch now to languages, and has a number of 38 languages. Is simpler for me to just read that number off from the Google Analytics page than dig deep into it.

To me the numbers of country counts are down a bit. I know not why.

But then again, how does it matter across the board? Well reality is means I have a direct connect which exists by objective measures. I'm just reading off information conveniently provided by the vast resources of Google.

Will there be a shift for me based on these numbers? Probably not because of them but because of changes in my use of the blog as is no longer primarily a research blog as have stopped new mathematical research. That freed up a lot of mental energy for other things. Math research is fun yes, but EXHAUSTING.

Now it's more interesting for me to consider what I've learned. And yes, continue to learn, but not chasing possibility wherever the math might take me.

Now it's more like sitting with the math and chatting conversationally. Which is a lot more relaxing.

James Harris

Sunday, December 04, 2016

When simplicity bemuses

One problem I have had for two decades now is when I look at something that seems too simple to me that I discovered. Which is not just a problem I have as get a weird pleasure out of noting that my first paper sent off to a math journal was covering packing of spheres and was rejected as too simple. Here's a link to a post I finally put on the blog in 2008 discussing, and yeah I lost the paper. So I had to recollect the argument.

Far as I know the original paper which was on paper is gone. Probably tossed it into the trash or something, which doesn't matter. My problem with it was, how could there be this simple approach to a problem that was over two thousand years old?

I'm just not able to maintain confidence on that one as I tell myself, but Sir Isaac Newton worked on this problem, how could he not have noticed this simple approach? Then am like, but I don't need it anyway. I say that about lots of things. Is like, who cares, and I look at something else I have. And now? Have TONS.

Like take a look at this one. Copying over from a post on my Beyond Mundane blog though LOTS of posts about it on this one:

With natural numbers--means use ints or longs--where pj is the jth prime:

P(x,n) = x - 1 - sum for j=1 to n of {P(x/pj,j-1) - (j-1)}

It counts primes when n equals the count of primes up to sqrt(x), so if n is greater than the count of primes up to and including sqrt(x) then n is reset to that count.

There is nothing simpler which is as fast for counting prime numbers in ALL of mathematics. And nothing faster for its size. But is SO simple. For years will admit bugged myself by wondering why Archimedes doesn't have it, and how would human history have changed if he had?

And that's not even expressing it pretty as Blogger doesn't make that easy far as I know. If it IS doable without a bunch of extras would like to know! Would switch to showing that way. Is MUCH prettier with the summation sign used.

These days am not as bemused about it as in the past, but still am to an extent I rarely admit. It uses mathematics that is easy, as you really only need division. And in a more general form you don't even have to know what a prime is, as it will select them out.

The math knows what a prime number is. There is nothing else like it in all of mathematics and it even leads to a partial differential equation.

That the math does know what a prime number is without some human guiding it used to mess with my mind massively. And I realized, the math does not need us.

So why is such a simple discovery mine? I could go on. I could fill this post with simple results that bemuse me. I even improved upon work by Gauss. Found results in areas where Euler and Ramanujan both considered something. And what of it?

When I need a pick me up though can just do Google searches. And even at times will make myself look for math things done by others related to my results. I wish they were as interesting though. That's better for the human species.

One thing will admit, I don't know how anyone who knows mathematical history thinks a person has just one result. Or even a couple and is a big deal. You want to compete in mathematical discovery? Start talking with at least half a dozen major results, maybe. It's an incredibly competitive arena. Maybe most competitive of all human intellectual endeavors.

I shrug at people claiming a single result. Am like, maybe you're getting started? But probably not or you'd know better. Come back when you have a dozen.

And have thought now for years about how these are mine, but reality is I'm just some guy who figured out some math.

Maybe the reality is we really will never know why about some thing.

But how do you know?

James Harris