Saturday, April 04, 2009
JSH: Weird blind spots with Pell's Equation
Looking over the literature over Pell's Equation available on the web, it appears that there are weird blind spots in what is commonly reported related to results having to do with:
(x-1)(x+1) = Dy^2
leading to some simple relations for x, which give you what I call the alternates to Pell's Equation:
j^2 - Dk^2 = -1, j^2 - Dk^2 = -2, and j^2 - Dk^2 = 2
The alternates are easier to solve in general because j is roughly sqrt(x), and gives you x, where the first j that works gives you the first x, so there is NO good reason to work on Pell's Equation directly in general. But things are weirder than that!!!
There are some easy solutions for special cases of D. For instance, if D = n^2 - 2, then the first x = D+1.
So there's no point in records with Pell's Equation as you can just take any integer n, square it, subtract 2, and for that D, you have the first x from D+1.
But also you have a result about n^2 - 2, as it will tend to be prime as it must be a prime or the product of primes which match by quadratic residues for -1, -2, or 2.
That is, all prime factors of n^2 - 2, must ALL have either -1, -2, or 2 as a quadratic residue!
My guess on what may have occurred is that with the continuing fractions solution in hand, math people don't just play with the full equation, and are taught to just solve Pell's Equation using continuing fractions or something.
Euler and Fermat may not have cared to talk as much about the alternates for their own weird reasons.
Remember they were involved in challenges and stuff. It was an intellectual activity for them which was somewhat about passing the time which could be a big deal for members of their class.
And Fermat was a lawyer. For him math was just that thing he did out of boredom (or he was driven for reasons he didn't fully understand).
Like can you imagine? He was shown the rational parameterization of Pell's Equation but couldn't be bothered with it. Yet conics are a HUGE area, and rational parameterizations have practical application.
The oddities of some ego games by some great men long dead may have skewed history in a fascinating way.
(x-1)(x+1) = Dy^2
leading to some simple relations for x, which give you what I call the alternates to Pell's Equation:
j^2 - Dk^2 = -1, j^2 - Dk^2 = -2, and j^2 - Dk^2 = 2
The alternates are easier to solve in general because j is roughly sqrt(x), and gives you x, where the first j that works gives you the first x, so there is NO good reason to work on Pell's Equation directly in general. But things are weirder than that!!!
There are some easy solutions for special cases of D. For instance, if D = n^2 - 2, then the first x = D+1.
So there's no point in records with Pell's Equation as you can just take any integer n, square it, subtract 2, and for that D, you have the first x from D+1.
But also you have a result about n^2 - 2, as it will tend to be prime as it must be a prime or the product of primes which match by quadratic residues for -1, -2, or 2.
That is, all prime factors of n^2 - 2, must ALL have either -1, -2, or 2 as a quadratic residue!
My guess on what may have occurred is that with the continuing fractions solution in hand, math people don't just play with the full equation, and are taught to just solve Pell's Equation using continuing fractions or something.
Euler and Fermat may not have cared to talk as much about the alternates for their own weird reasons.
Remember they were involved in challenges and stuff. It was an intellectual activity for them which was somewhat about passing the time which could be a big deal for members of their class.
And Fermat was a lawyer. For him math was just that thing he did out of boredom (or he was driven for reasons he didn't fully understand).
Like can you imagine? He was shown the rational parameterization of Pell's Equation but couldn't be bothered with it. Yet conics are a HUGE area, and rational parameterizations have practical application.
The oddities of some ego games by some great men long dead may have skewed history in a fascinating way.
# posted by James Harris @ 20:53 
