Mystery with Pell's Equation parametric solution

What I've managed to find from a result that followed from my Quadratic Diophantine Theorem is a parametric solution to Pell's Equation in rationals:

With x² - Dy² = 1

I have proven:

y = 2[f₂v - 1]/[D - (f₂v - 1)²]

and

x = [D + (f₂v - 1)²]/[D - (f₂v - 1)²]

where f₁f₂ = D-1, and the f's are non-zero integer factors, while v is nonzero but is otherwise a free variable.

Which is, of course, looks similar to the parametric solution for circles in rationals discovered in antiquity:

x = (1 - t²)/(1 + t²)

y = 2t/(1 + t²)

See: http://mathworld.wolfram.com/Circle.html eqns. 16 & 17

It looks similar as it is the parametric solution for circles when D=-1, for ellipses with D<0, and of course, hyperbolas are given by D>0.

In considering what I see as the mystery of the quietness in the mathematical literature around this solution, it seems possible that the focus on Pell's Equation as a Diophantine equation, considering integers only, can be part of it.

Research on the web indicates that the parametric solution I've found was known much earlier, as Fermat was aware of it. So now I wonder why it gets so little mention and why wasn't the relation to rational parametric solutions for circles, ellipses and hyperbolas noted? Or was it? I'm continuing research.

There are other interesting things I've found.

For example, for any positive integer D, if D+2 is a perfect square, then D+1 is the first solution to Pell's Equation for x.

e.g. 8² - 7*3² = 1, 15² - 14*4² = 1

In general, if D+2 is a perfect square, then D = n² - 2 for some natural number n, and

(n² - 1)² - (n² - 2)n² = 1

and notice n⁴ - 2n² + 1 - n⁴ + 2n² = 1.

But that is probably something so well-known that it's considered trivial.

Another thing is the results with the negative Pell's Equation and two other alternates shows that in general the alternates can be solved in order to solve Pell's Equation, as they must exist for prime D, and a method can be generalized for composite D as well.

e.g. For the famous D=61 case, the solution from the negative Pell's Equation is not as spectacular:

j² - Dk² = -1

29718² - 61*3805² = -1

gives x = 1766319049, from x = 2j² + 1.

1766319049² - 61*226153980² = 1

The difference in computation is staggering though as my example above suggests, as the first x for Pell's Equation is roughly the square of the first j for the alternates in all cases, and the same techniques that work for Pell's Equation work for the alternates, for instance, continued fractions.

The issue is I'm guessing not minor, as consider a paper I found with some digging on the web from Los Alamos National Laboratory researchers:

...Zeros of weight 1 3j- and 6j-coefficients are known to be related to the solutions of classic Diophantine equations. Here it is shown how solutions of the quadratic Diophantine equation known as Pell's equation are related to weight 2 zeros of 3j- and 6j-coefficients.

Link to full citation


James Harris

Advancing tautological spaces methodology

It has been almost nine years since I came up with the idea I now call using tautological spaces as that happened back in December 1999, and since that time I've done some work on terminology (link is a reference for the rest of this page).

Now after my success with a general method to reduce binary quadratic Diophantine equations, I see a need to advance the methodology of tautological spaces where that is simple enough.

To handle quadratic Diophantines in general I see a need for a more general tautological space:

a₁^h + a₂^h +v₁a₃^h +...+v_n-2a_n^h = 0(mod a₁^h + a₂^h +v₁a₃^h +...+v_n-2a_n^h )

to handle conditionals with n variables, where I call h, the hyperdimension.

As an example consider n=4, where all the conditional variables are raised to the 1st power so h=1, as then the tautological space is described by:

T_S(1,1,{1,1},{1,1})

So it is a 6-dimensional tautological space a hyperdimension of 1, so there are two free variables which by my type of usage would be v₁ and v₂, so that looks like:

x + y + v₁z + v₂w = 0(mod x + y + v₁z + v₂w)

The v's give additional degrees of freedom to allow reducing out to a general solution like with my Quadratic Diophantine Theorem.

Actually I realized the need to extend tautological space terminology by wondering about considering solutions where there are more conditional variables than say, x, y and z.


James Harris

Edited 6/24/12 to correct leaving off hyperdimension which IS necessary.

All about identities

The role of identities in mathematics is so familiar that most people take it for granted while my findings show that identities can be seen in an entirely new light.

Identities are simple enough as they are always true and are the tautologies of mathematics.

For instance 1=1 and x=x are both identities.

Mathematics uses identities all the time as consider the very easy solution of the following equation:

x-2 = 0

as you can use the identity 2=2 to find that

x-2+2 =2, so x=2

and you have a solution that IS so trivial that most probably don't think of identities when they solve such an equation.

A slightly more complicated but still easy use of identities is with completion of the square, as for instance, given

x² + 2xy = z²

adding the identity y² = y² gives you

x² + 2xy + y² = y² + z²

which of course is just (x+y)² = y² + z², allowing you to take the square root of both sides and solve to find

x = -y +/- sqrt(y² + z²).

Despite their importance in algebra where you cannot really do algebra without identities, development of the algebra of identities has not been extensive up until now which can be seen by comparing from my own research where I rely on more complicated identities still like

x² + y² + vz² = x² + y² + vz²

where to a large extent what I do is like what was shown before, except that the identities that I add end up being a lot more complicated, and are being used for more complex results.

That brings up the need for classification, where I've decided that the word "identity" doesn't do enough to cover everything so I introduced the phrase "tautological space", as with my use of identities the identity is not an afterthought at the end just to get to the answer, but a start of the argument and a dominant part of the proof.

Continuing now to classify all identities as tautological spaces, I've used the number of variables to give the dimension, so that with 4 variables you have a 4-dimensional tautological space.

The simplest identities like 2=2 that have no variables, have no dimension and I call them unary tautological spaces because they are all multiples of 1=1.

If any variables of the tautological space are raised to a power other than 1 then that is a hyperdimension and is shown by a set of natural numbers. Variables in a tautological space cannot be raised to a negative power, nor to 0 as that is redundant since instead a constant can be used.

For example with x² + y² + vz² = x² + y² + vz² you have a 4-dimensional tautological space with a hyperdimensional set of {2,2,1,2}.

Since the hyperdimensional set gives the number of dimensions, more compactly then you can describe a tautological space as follows.

T_S{2,2,1,2}

And that allows a complete classification of all possible identities.

Coming back to look this over I see that it does not, so I'm updating with a change in terminology.

The free variable which I traditionally call v is given by the 1, but I think I can use nesting to show where it is:

T_S(2,2,{1,2})

So that means a 4-dimensional tautological space where the conditional variables are all squared.


James Harris

Rational parameterization conic sections result

It's worth pulling out into a separate post a wonderful unifying result for 3 of the 4 conic sections that follows from the parameterization of Pell's Equation:

Given x² - Dy² = 1, in rationals:

y = 2t/(D - t²)

and

x = (D + t²)/(D - t²)

showing it more traditionally versus the way that results from my own derivation.

You get hyperbolas with D>0, the circle with D=-1, and ellipses with D<0.

You can see the D=-1 case from a well-known mainstream source at the following link:
See: http://mathworld.wolfram.com/Circle.html eqns. 16 & 17

I actually feel very honored to have my own derivation leading to such a beautiful result, which allows categorization of ellipses and hyperbolas by a single number, which then is probably connected to the eccentricity.

So what mathematicians call "Pell's Equation" is in other areas a one-stop equation for generating 3 of the 4 conic sections in rationals just by fiddling with one variable--D.


James Harris