### Thursday, September 25, 2008

## JSH: Diophantine chains and x^2 - 2y^2 = 1

Given the general result that whenever you have

x^2 + Dy^2 = F

you have a connected equation

z^2 + D(x+y)^2 = F*(D+1)

where finding a solution for it gives you a solution for the first, you can immediately figure out how to get an infinite number of solutions with D=-2 and F=1, the classical Pell's equation, as then you have

x^2 - 2y^2 = 1, followed by

z^2 - 2(x+y)^2 = -1

and the next in the series is

w^2 - 2(x+y+z)^2 = 1

so you just get this flipping back and forth, and with one solution at the start you can get the solutions that follow, so with x=3, and y=2, you have next that

z^2 - 2(5)^2 = -1, so z^2 = 49, so z=7, and then you have

w^2 - 2(3+2+7)^2 = 1, so w^2 - 2(144) = 1, so w^2 = 289 and w=17.

And you can do that forever.

It's kind of weird to look at an elementary result that gives all the answers where other techniques were used before, and consider how that relates to them.

But even weirder that I have the general result that is part of a theory that handles ALL Diophantine quadratic equations in 2 variables, except for D=-1, which just gives you a factorization result and no chain.

x^2 + Dy^2 = F

connecting to

z^2 + D(x+y)^2 = F*(D+1)

is the underlying machinery which leads to an elementary solution, and now one can ponder how that simple answer relates to the more complicated ones in the area, like in explaining exactly what is really happening with continued fractions—regardless of the classical explanation.

After all, centuries ago men believed that the planets moved in circular orbits because they thought that was perfection and why would God do it any other way? But perfection turned out to be elliptical orbits and beyond.

Advancements in knowledge advance explanations—from the primitive to the new as our knowledge as a species, grows over time.

x^2 + Dy^2 = F

you have a connected equation

z^2 + D(x+y)^2 = F*(D+1)

where finding a solution for it gives you a solution for the first, you can immediately figure out how to get an infinite number of solutions with D=-2 and F=1, the classical Pell's equation, as then you have

x^2 - 2y^2 = 1, followed by

z^2 - 2(x+y)^2 = -1

and the next in the series is

w^2 - 2(x+y+z)^2 = 1

so you just get this flipping back and forth, and with one solution at the start you can get the solutions that follow, so with x=3, and y=2, you have next that

z^2 - 2(5)^2 = -1, so z^2 = 49, so z=7, and then you have

w^2 - 2(3+2+7)^2 = 1, so w^2 - 2(144) = 1, so w^2 = 289 and w=17.

And you can do that forever.

It's kind of weird to look at an elementary result that gives all the answers where other techniques were used before, and consider how that relates to them.

But even weirder that I have the general result that is part of a theory that handles ALL Diophantine quadratic equations in 2 variables, except for D=-1, which just gives you a factorization result and no chain.

x^2 + Dy^2 = F

connecting to

z^2 + D(x+y)^2 = F*(D+1)

is the underlying machinery which leads to an elementary solution, and now one can ponder how that simple answer relates to the more complicated ones in the area, like in explaining exactly what is really happening with continued fractions—regardless of the classical explanation.

After all, centuries ago men believed that the planets moved in circular orbits because they thought that was perfection and why would God do it any other way? But perfection turned out to be elliptical orbits and beyond.

Advancements in knowledge advance explanations—from the primitive to the new as our knowledge as a species, grows over time.