Wednesday, March 25, 2009


Pell's Equation simplified

I've proven that given:

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]


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 equations may seem to some like just a curiosity but you can do figure out why Pell's Equation behaves as it does using them.

One result I just got from them is that if p+2 is a square, then if D = p, Pell's Equation has x = p+1 as a solution.

For instance, for D=7, x=8, works, as 8^2 - 7(9) = 1, and for D=23, x=24, works as 24^2 - 23^2(25) = 1.

The behavior of Pell's Equation is about factoring. Everything has to do with how D itself factors.

The equations are remarkably simple given the history around Pell's Equation.

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