### Wednesday, September 16, 2009

## JSH: Binary quadratic Diophantines

Last year I found a nifty mathematical short-cut for handling with integers equations of the form

c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y

which are called binary quadratic Diophantine equations, where x and y are unknown and the c's are known constants, and my understanding is that they come up in quantum mechanics.

There ARE various methods for getting from the general equation to an equation of the form u^2 + Dv^2 = F, which I've seen called the general Pell's Equation, where solving it allows you to solve the original, but I found the fastest short-cut out there:

(2A(x+y) - B)^2 - 4As^2 = B^2 - 4AC

where A, B and C are defined by the c's and are:

A = (c_2 - 2c_1)^2 + 4c_1(c_2 - c_1 - c_3)

B = 2(c_2 - 2c1)(c_6 - c_5) + 4c_5(c_2 - c_1 - c_3)

and

C = (c_6 - c_5)^2 - 4c_4(c_2 - c_1 - c_3).

The s is actually I think x-y, but I usually just give it as s, as it's still just another unknown.

That IS the short-cut approach to generally handling binary quadratic Diophantine equations which has only one special case, when c_2 = 2c_1, and c_2 - c_1 - c_3 = 0, which is handled with a substitution, like y = x + z, and then it all works, as otherwise you get 0 = 0.

Now I discovered that mathematical tool last year and as I'm proud of it, I've talked about it and I've sent it to mathematicians and even tried to get a paper published, to no avail.

So I'm back on Usenet trying to reach the physics community directly, as it IS a tool. And yes, there are Usenet posters who make it their business to insult people but that's because that's what they do—insult people.

There is no other method that works as directly as a short-cut for binary quadratic Diophantine equations and it is a shame that academics around the world have built a system that allows someone like me to be blocked possibly just because I don't have a math Ph.D, or any Ph.D and you all know that's a really close club.

Math Ph.D's might piss on themselves rather than admit that someone without even a math degree, as my undergrad is in physics, might figure out something really BIG.

Maybe they're afraid it'd be the end of the world as they know it.

Regardless, they've fought me before to a standstill and killed a math journal to do it. Google: SWJPAM

I've countered by putting information out there and pointing out when Google pulls the world to it, like do this search in Google.

Search string: binary quadratic Diophantine equations

I call these the Math Wars. I discover things, math people ignore my discoveries, while some fringe Usenet freaks insult me.

The Math Wars are about math society as an institution foolishly believing that in its numbers and influence it can stop the research of one man, me.

We are battling to see if they are right. And the world feels the heat of that massive exchange.

You can side with the mathematicians in the Math Wars if you wish, and use weaker techniques that are not as fast or as effective, or you can advance the human species.

Mathematicians are mostly Ph.D's who are too small-minded to understand that human Progress is bigger than even their massive egos, and sure, maybe they can block my research for a few more years, but when it breaks through, they diminish all academics, as if mathematicians can ignore simple mathematics easily proven, why can't ANY academic ignore the truth?

Why should the world then trust academia?

Why not just shatter it entire and start over?

The Math Wars can only last so long. For me to lose this research must be lost…

And humanity must then work harder than it need, and do less than it can, so some petty people can protect their turf.

c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y

which are called binary quadratic Diophantine equations, where x and y are unknown and the c's are known constants, and my understanding is that they come up in quantum mechanics.

There ARE various methods for getting from the general equation to an equation of the form u^2 + Dv^2 = F, which I've seen called the general Pell's Equation, where solving it allows you to solve the original, but I found the fastest short-cut out there:

(2A(x+y) - B)^2 - 4As^2 = B^2 - 4AC

where A, B and C are defined by the c's and are:

A = (c_2 - 2c_1)^2 + 4c_1(c_2 - c_1 - c_3)

B = 2(c_2 - 2c1)(c_6 - c_5) + 4c_5(c_2 - c_1 - c_3)

and

C = (c_6 - c_5)^2 - 4c_4(c_2 - c_1 - c_3).

The s is actually I think x-y, but I usually just give it as s, as it's still just another unknown.

That IS the short-cut approach to generally handling binary quadratic Diophantine equations which has only one special case, when c_2 = 2c_1, and c_2 - c_1 - c_3 = 0, which is handled with a substitution, like y = x + z, and then it all works, as otherwise you get 0 = 0.

Now I discovered that mathematical tool last year and as I'm proud of it, I've talked about it and I've sent it to mathematicians and even tried to get a paper published, to no avail.

So I'm back on Usenet trying to reach the physics community directly, as it IS a tool. And yes, there are Usenet posters who make it their business to insult people but that's because that's what they do—insult people.

There is no other method that works as directly as a short-cut for binary quadratic Diophantine equations and it is a shame that academics around the world have built a system that allows someone like me to be blocked possibly just because I don't have a math Ph.D, or any Ph.D and you all know that's a really close club.

Math Ph.D's might piss on themselves rather than admit that someone without even a math degree, as my undergrad is in physics, might figure out something really BIG.

Maybe they're afraid it'd be the end of the world as they know it.

Regardless, they've fought me before to a standstill and killed a math journal to do it. Google: SWJPAM

I've countered by putting information out there and pointing out when Google pulls the world to it, like do this search in Google.

Search string: binary quadratic Diophantine equations

I call these the Math Wars. I discover things, math people ignore my discoveries, while some fringe Usenet freaks insult me.

The Math Wars are about math society as an institution foolishly believing that in its numbers and influence it can stop the research of one man, me.

We are battling to see if they are right. And the world feels the heat of that massive exchange.

You can side with the mathematicians in the Math Wars if you wish, and use weaker techniques that are not as fast or as effective, or you can advance the human species.

Mathematicians are mostly Ph.D's who are too small-minded to understand that human Progress is bigger than even their massive egos, and sure, maybe they can block my research for a few more years, but when it breaks through, they diminish all academics, as if mathematicians can ignore simple mathematics easily proven, why can't ANY academic ignore the truth?

Why should the world then trust academia?

Why not just shatter it entire and start over?

The Math Wars can only last so long. For me to lose this research must be lost…

And humanity must then work harder than it need, and do less than it can, so some petty people can protect their turf.