Tuesday, February 24, 2009
JSH: Welcome to the real world
With some very easy algebra I can prove to those of you who are intellectually honest, how little you understand of how knowledge is accepted today.
I found a connection between the equation commonly called Pell's equation
x^2 - Dy^2 = 1
and an expression which I like to say is a discrete ellipse or it is a Pythagorean Triple with integer x and y, if D-1 is a square:
(D-1)j^2 + (j+/-1)^2 = (x+y)^2
where j = ((x+Dy) -/+1)/D.
That expression connects rational solutions of the first equation to rational solutions of the second, which may not seem like a big deal until you see that a factorization of the D in the first is connected to a factorization of D-1 in the second:
(x-1)(x+1) = Dy^2
and
(D-1)j^2 = (x+y + (j+/-1))(x+y - (j+/-1))
and it doesn't take long to wonder, if you think about it, if you can factor efficiently using that remarkable relation.
That's the easy part. Convincing people that the factoring problem could go down so easily—now that is hard.
I've done more to explain in other posts so I'll step away from that to emphasize to you: mathematical proof, easy algebra, major importance in the factoring problem.
And let you ponder your own actions which I assume are to do nothing.
(If you are doing something, YEAH!!! This post does not apply to you.)
We live in a structured world with a LOT of assumptions. People around the world are already seeing many of those assumptions shattered with the financial crisis.
I'm hoping some of you read this post, and if you do nothing, you go to bed tonight, and tomorrow think about why you do nothing.
And then understand HOW we have financial crises and so many other crises in our world as you realize how hard it can be, to accept absolute mathematical proof.
Absolute mathematical proof.
Now think about what happens when people don't even have that, and your world makes a lot more sense.
It is harder than most people realize to do the right thing, even when you have absolute knowledge of what the truth must be.
I found a connection between the equation commonly called Pell's equation
x^2 - Dy^2 = 1
and an expression which I like to say is a discrete ellipse or it is a Pythagorean Triple with integer x and y, if D-1 is a square:
(D-1)j^2 + (j+/-1)^2 = (x+y)^2
where j = ((x+Dy) -/+1)/D.
That expression connects rational solutions of the first equation to rational solutions of the second, which may not seem like a big deal until you see that a factorization of the D in the first is connected to a factorization of D-1 in the second:
(x-1)(x+1) = Dy^2
and
(D-1)j^2 = (x+y + (j+/-1))(x+y - (j+/-1))
and it doesn't take long to wonder, if you think about it, if you can factor efficiently using that remarkable relation.
That's the easy part. Convincing people that the factoring problem could go down so easily—now that is hard.
I've done more to explain in other posts so I'll step away from that to emphasize to you: mathematical proof, easy algebra, major importance in the factoring problem.
And let you ponder your own actions which I assume are to do nothing.
(If you are doing something, YEAH!!! This post does not apply to you.)
We live in a structured world with a LOT of assumptions. People around the world are already seeing many of those assumptions shattered with the financial crisis.
I'm hoping some of you read this post, and if you do nothing, you go to bed tonight, and tomorrow think about why you do nothing.
And then understand HOW we have financial crises and so many other crises in our world as you realize how hard it can be, to accept absolute mathematical proof.
Absolute mathematical proof.
Now think about what happens when people don't even have that, and your world makes a lot more sense.
It is harder than most people realize to do the right thing, even when you have absolute knowledge of what the truth must be.
# posted by James Harris @ 21:50 
