Tuesday, January 15, 2008

 

JSH: Questioning certainty, again

I have to apologize as I came forward with a perfect mathematical argument and let myself be convinced that there were flaws and put forward "fixes" that were wrong.

The story is that a poster claimed that with my congruence relations T=21 did not work, and I'd had problems myself with a reversal of the method to resolve quadratic residues, with, yup, 21, so I was wondering if for some reason 3 was a problem for the relations.

So when he claimed that the congruence relations didn't work for 21, I kind of leapt to a conclusion, and started adding on additional conditions i.e. that nT be coprime to 3, and odd.

But they are not necessary.

It is an odd thing in this position with a dramatic argument in a well-worked area where I had a simple idea and saw it through to some relatively simple mathematics which reduces to a couple of easy Diophantine equations.

And yes, they ARE easy, which is how I realized that I'd added on unnecessary conditions by just solving out integer solutions with both equations with nT=21.

Now then, there is still one major issue where I must admit to being fascinated by human nature.

The derivation of my factoring congruence relations is all about figuring out z mod p, and y mod p, and I don't tell the algebra to mostly worry about nT mod p. It does that on its own.

But in doing so, it's saying that as far as the math is concerned that is all the information that is needed!!!

That was the other area where I questioned certainty—and here certainty is about mathematical proof.

What that means is that if you have an alpha, and you have a k, then you have a corresponding integer solution to z^2 = y^2 + nT, where y and z have the given residue modulo p.

That is just so freaking weird. It still kind of bugs me, as it's just so freaking weird.

So the congruence relations do solve the factoring problem if you resolve them with p>sqrt(T), and n=1, and get z-y mod p and z+y mod p as one of them will give you the smaller prime, directly.

It kind of bugs me though.

So I'll digress: George W. Bush invaded Iraq on false pretenses when the the now dead, but previously then alive Iraqi government actually offered to let CIA agents come into the country and go to areas where supposedly "weapons of mass destruction" existed. During the "shock and awe" part of the US campaign one of the news stories that distinguished itself in my mind was the accidental bombing of a ward of pregnant women in Iraq.

Oddly enough, at about the same time in the US, a dog fell through the ice on a partially frozen pond, and the country erupted with an outpouring of concern and relief when the dog was saved by brave rescuers.

And I realized that most human beings are not only not logical, they aren't really in the real world, but live in a world of their own perceptions.

Mathematics is for logical people.

And that is why I have problems getting my results acknowledged.

Most of you, are not logical people. You are people who when told will cheer the saving of a dog, and ignore the killing of pregnant women and babies, if that is what you are told to do, even if you are not in the United States. You are the same people. I know this because I see how you react to the truth, whether it's mathematical or in the "real world".

To most of you, your perception is reality, and you do not care—unless it's your wife and kid in that pregnancy ward.





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