Monday, April 24, 2006


JSH: Two mysteries, quadratic residues still

Looking over these nifty results I now have with quadratic residues I am amazed, yet again, with simple research results available in yet another high profile area.

It is amazing to me that I just kind of think about some area for a little while and out pops a remarkable simple answer, and then there is the question of why didn't anybody else think of it before?

With prime residues I have two mysteries, so you can see how I consider these kinds of questions as I work to understand how centuries could pass without people figuring out things I just get, often with little effort.

With quadratic residues, people have played with them a lot for years and n^2 - r is not this new thing. Neither is the reality that it has a masking effect for r not a square, where I'm only interested in natural numbers, of course.

So why wouldn't people have routinely used that masking effect of quadratic residues to find primes?

My quick answer is, they didn't have computers.

Gauss didn't want to just get big primes. He'd want to do things like get all primes up to some value, or count primes.

They did want to know about relations of quadratic residues to each other, and Euler did a lot of work in that area, but did any of them care about questions like, how many primes up to 1000 have 2 as a quadratic residue?

Today, computers make it easy to ask lots of simple questions about quadratic residues and write programs that just go look. Even if you don't know a lot about the subject, modern computers are powerful enough that brute force will get you a lot of answers.

In the past, they had pen and paper, patience, and the will to know.

That's one mystery, but the other is odder as my simple result relating quadratic residues to Goldbach's conjecture did not require computers.

Playing with it does not require computers as you can look at p_1 + p_2 and consider this relationship with the factorization of 2(p_1 + p_2) and it's just this neat thing.

I am reminded of how I've often wondered about my prime counting function, and why wouldn't past mathematicians have figured it out?

The derivation is straightforward, and to me rather easy, though, as I've pointed out, no other human being that I've seen has been able to step through from beginning to end a similar derivation, even after looking at mine.

In any event, the Goldbach relationship is what is probably going to force all of this into the public eye, and maybe math historians can help then on the questions about how much was known before by my predecessors, and how they could have missed things that I so easily catch.

Oh yeah, so now you have some sense of why it is so difficult to handle a major discoverer.

History books don't teach you these things, I guess.

The problem is that you can't know what I'll figure out next, and I just keep figuring things out.

That's how people like me create history. We are so prolific that others can work for centuries on doors we open.

People trying to block my work would probably prefer me to just keep talking about FLT, which, you'll notice, is all that's even mentioned on when it attacks me.

Some of you may focus on my research about non-polynomial factorization, working to try and block it.

While others are dismissive of my factoring approaches and terminology like "surrogate factoring".

Oh, and of course, there are those of you who actively work to block interest in my prime counting function, and the partial differential equation that follows from it, which could lead to proving or disproving the Riemann Hypothesis.

But now I'm going into quadratic residues and the Goldbach conjecture, and possibly it's sinking in for some of you now, you cannot win.

People like you may have been after Euler or Gauss, but history doesn't record it, so you don't understand the lessons of history, but I do.

I think this time it should be fully documented that people like you exist, that Newton wasn't just this nasty person, but likely faced all kinds of things, from people like you, that history didn't record, and that part of the problem—why people like you go on for as long as you do—is that you don't have worries about consequences.

I think now is the time to change that so that the lesson is indelible in history.

Yes, we always beat you.

Hmmm…yes, we do always beat you. Maybe the best lesson from me here is that winning wasn't in doubt, but learning about who I am, and what I must do, were.

I needed the challenge, I guess. I had to fight through, not just in discovering what I did, but in learning about humanity, and why it all is important.

It couldn't be easy.

It never is, for one of us.

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