Harristotelian Logic

Rather remarkably I think I have found a way to generally solve for Pell's Equation, which brings up the issue yet again of "hidden variables", and raises severe implications ranging from mathematical viewpoints all the way to Internet security (possibly unfortunately).

First it helps to see the result!!!

General solution to Pell's Equation

Given

x^2 - Dy^2 = 1

I have proven:

y = [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

and

x = +/-(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v) - [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

where f_1*f_2 = D-1, and the f's are non-zero integer factors, while v is a free variable.

As Pell's Equation is normally considered in integers as a Diophantine equation note that you find rational v such that x and y are integers, which gives the 'why' of Pell's Equation. For instance, for D=2, f_1*f_2 = 1, so I have:

y = [+/-4v/(1 - v^2 - 2v) +/- 1 -/+(1 + v^2)/(1 - v^2 - 2v)]

and

x = +/-(1 + v^2)/(1 - v^2 - 2v) - [+/-4v/(1 - v^2 - 2v) +/- 1 -/+(1 + v^2)/(1 - v^2 - 2v)]

where I notice an easy case to give integer solutions with v = -2 (but v can be a fraction as well so that is just what worked for this example), as I have then:

y = [-/+8/(1 - 4 + 4) +/- 1 -/+(1 + 4)/(1 - 4 + 4)] =[-/+8 +/- 1 -/+ 5]

and

x = +/-(1 + 4)/(1 - 4 + 4) - [-/+8/(1 - 4 + 4) +/- 1 -/+(1 + 4)/(1 - 4 + 4)] = +/-5 - [-/+8 +/- 1 -/+ 5]

So I get several solutions with that choice.

For instance, y = 8 - 1 + 5 = 12 is a solution, with x = 5 + 12 = 17, as 17^2 - 2(12)^2 = 1.

Derivation of the result was done using research found by doors opened by my Quadratic Diophantine Theorem, my use of it against Pell's Equation, and my research result linking Pell's Equation to discrete ellipses and Pythagorean Triples.

The weird thing here then is that you can just solve for x and y in Pell's Equation in terms of an additional variable, which is otherwise nowhere to be found in x^2 - Dy^2 = 1. So an additional previously hidden variable allows functional solutions that encompass all behavior of Pell's Equation, whether people realize it is there or not.

But now x and y are functions of v, so I can have x = r(v)/t(v), and y = s(v)/t(v), so that I can solve to find:

(r(v) - t(v))(r(v) + t(v)) = D(s(v))^2

and remarkably enough, turn factoring into a calculus problem of finding minima when D is a composite you are trying to factor!!!

So the reality of the result has HUGE implications in lots of areas, and it should be of interest I'd think, how I even found the solution.

I've previously posted today on some math newsgroups to get some independent verification which I've gotten so the equations are correct.

That changes the game. We can solve problems in new ways. And in so doing get direct answers even in areas where people thought they had all the answers, as I note that continued fractions have been considered the way to go with Pell's Equation, while this research opens the door to an entirely new way to solve with reasons for why it can be more difficult for certain values of D that continued fractions could NOT explain.

It answers the 'why' of Pell's Equation—by introducing an additional previously hidden variable dependency.

Quite simply it is a revolutionary discovery as to how to approach certain types of problems.