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Wednesday, July 22, 2015

Reality check with Ramanujan

Last year I was gratified to see I could expand upon something noticed by Ramanujan.

He noticed that 2n - 7 = x2 had integer solutions for 5 values of n, and you can read up on that at MathWorld. And Euler had considered something more general. While I could generate those solutions with my own research.

Using my research I can show how to find integer solutions for x and y, when:

x2 + 7y2 = 2n

where with Euler's result, n is 3 or greater, and x and y are odd integers. He also found a really cool solution for x and y, while with my research you can generate solutions with my BQD Iterator.

The applicable Binary Quadratic Diophantine iterator here, or BQD Iterator for short is:

u2 + 7v2 = F

means that

[(u-7v)/2]2 + 7[(u+v)/2]2 = 2*F

where the iterator has been adjusted by dividing 4 off, which requires u and v odd integers. Letting u = v = 1, gives F = 8, and you get Euler's result, as the iterator goes out to infinity.

So you just plug the number in and watch it go. I actually did that with a prior post last year when I discussed these things as I found them.

For me it's a maybe minor result which is so awesomely important because it relates to Ramanujan and Euler. Which is the kind of thing that is remarkable for someone. I wonder how many people in history will ever be able to explain something beyond what was known by those two when they were working on the same problem?

Is there anyone else in mathematics?

And there you go. It's a reality check with Ramanujan. And I guess Euler but that would make too long of a blog post title.

I have no clue how many people have managed such a thing, but as one of them I'll admit that it was a confidence booster. And it was me heading off in that direction after working at generating solutions for equations with Mersenne numbers, and I just picked one to see, and found that Euler and Ramanujan had been intrigued by similar things.

To me that's just awesome.

Got a bit of a reaction last year as apparently that find zipped around the globe. Didn't notice? If so, that's not my problem. My duty was to present publicly, which I did, on this blog.

The mathematical community cheers mathematics, its discovery, and the profundity of a discipline that spans all of human civilization.

Sometimes you can get a reality check that points out the obvious. But for the discipline as a whole, its history speaks for itself.


James Harris
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