### Sunday, September 07, 2008

It is with great joy that I note there is a huge result on the generalization from Pell's Equation that posters figured out quickly in hindsight from my initial post giving

x^2 - Dy^2 = 1

with a natural number D, which requires that there exist some integer S such that

x+y = sqrt((S^2 + D - 1)/D)

and also I guess I may as well add that

x^2 + Dy^2 = 1

requires that

x+y = sqrt((D - S^2 + 1)/D)

immediately explaining a finite number of solution.

One poster should have a place in mathematical history for being the first to give the general equation for

Ax^2 - Dy^2 = c

with A and D natural numbers and c a non-zero integer and that could be you.

I quote his proof:

Define S = Ax + Dy. Then:
S^2 + (D - A)c
= A^2x^2 + 2ADxy + D^2y^2 + Dc - Ac
= A(Ax^2 - c) + 2ADxy + D(Dy^2 + c)
= A(Dy^2) + 2ADxy + D(Ax^2)
therefore
x + y = sqrt((S^2 + (D - A)c)/(AD))

He has his place in mathematical history now.

That could have been yours.

Some of you seem to think that protecting your math professors is protecting your own careers.

You are very wrong.

And blocking your place in history.

Search in Google on my Quadratic Diophantine Theorem, learn the proof, and the techniques I used.

Study my mathematical approach and you can re-write the textbooks across number theory.

You can be the great mathematicians of the future.

Or you can try to ignore my growing body of research results so you can muddle along for years to try to maybe make it as post-grads if you can even manage to work in the mathematical field, to wait for decades for the older mathematicians to finally decide to retire, grubbing for research dollars.

The choice is yours.

Make mathematical history.

Or labor in the shadows for the rest of your life for older men who had their time.

Now can be yours: History. Greatness.

Discovery.
I'm seeing indications that the value of the result isn't jumping out

The result is that given

Ax^2 - Dy^2 = c, there must exist integer S such that

x + y = sqrt((S^2 + (D - A)c)/(AD))

so now, you can explain existence for

Ax^2 - Dy^2 = c

where A and D are natural numbers and c is a non-zero integer, as you have immediately that

S^2 = Ac mod D and S^2 = -Dc mod A

so, for instance, Ac = 6 with D=7 cannot give integer solutions.

i.e. 3x^2 - 7y^2 = 2 has no integer solutions.

Further, when solutions are available, you can search for S modulo D and A.

It is a remarkably powerful little result for its utility and ease of proof, where Mr. Rodgers it appears used hindsight with a result I originally gave about Pell's Equation which is that given

x^2 - Dy^2 = 1

there must exist an integer S such that

x+y = sqrt((S^2 + D - 1)/D)

which I found using my Quadratic Diophantine Theorem.

So far though he seems uninterested in either credit or appreciation for the result, which I guess is kind of sad.

It shows where some in the current mathematical community are today.

My guess is he is worried that he has inadvertently supported my research when he was trying to give evidence dismissing it by showing how easy the derivation is.

So you can see the political behavior I decry with a man trying to block his OWN place in mathematical history.

I suggest that mathematicians allow him to do so, as the result follow easily enough from my Quadratic Diophantine Theorem anyway, and maybe there then is a lesson here, and a question:

What could drive a man to try and forgo a place in mathematical history out of fear that he might support my research?