Harristotelian Logic

With my factoring hopes using Pell's Equation having taken a huge battering as I finally simplified my general solution, I'm now looking for a silver lining, and feel like I've found it with the 'why' of Pell's Equation revealed from the simplified equations.

The simplified general solution to Pell's Equation:

In rationals, x^2 - Dy^2 = 1

you can solve for x and y explicitly in terms of the integer factors of D-1, and a rational independent variable v, with:

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

and

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

where f_1*f_2 = D-1 where v is non-zero.

The v's and factor of D-1 may seem redundant, but now it's possible to consider when Pell's Equation is an integer, and since v can be a fraction, let v = n/m, then:

n = (sqrt[D(B-1)/(B+1)]+1)m/f_2

Where B is some integer. Intriguingly it turns out that x = B, for a solution to Pell's Equation.

And that relation tells the 'why' of Pell's Equation, as notice from some examples:

For D=7, I have x=8, y=3, is a solution, as 8^2 - 7(9) = 1, so B=8, f_1*f_2 = 6.

n = (sqrt[7(7)/(9)]+1)m/f_2 = (7/3 + 1)m/f_2 = 10m/(3f_2).

Want something bigger?

For D = 29, x = 9801, and y = 1820 are solutions, so f_1*f_2 = 28, and B = 9801, so

n = (sqrt[29(9800)/(9802)]+1)m/f_2

so

n = (sqrt[29(8)(35^2)/((2)(29)(169)]+1)m/f_2

which is

n = (70/13+1)m/f_2 = 83m/(13f_2).

So you can see that for D=29, the math had to find a number that achieved two base conditions, for some integers j and k:

x = 2Dk^2 - 1, and x = 2j^2 + 1

Another case with primes you will see is:

x = Dk^2 - 1, and x = j^2 + 1

And intriguingly, those two cases mean that:

j^2 - Dk^2 = -1

for the latter case, while for the former:

j^2 - Dk^2 = - 2.

So now you know why the so-called negative Pell's Equation is not always true. If it's not true then the second equation with -2, is true.

And now you know trivially how to find a solution to the negative Pell's Equation when it exists, from a solution to Pell's Equation.

For over a thousand years people have not known why a particular solution worked, even though they knew how to find them.

These equations don't necessarily help you find solutions to Pell's Equation, but they tell you why a particular solution works.

So, for the first time in thousands of years of human history working with the so-called Pell's Equation, people can finally know why a particular solution is required.