[seqfan] Re: Pythagorean triples of triangular numbers?

[Brad Klee]

Hi Andy,

I'm not making any claims about homogeneous or projective coordinates.
My previous message only suggests the idea of taking a plane section of
what you
call an "affine surface". Curves in the (x,y) plane have square dihedral
symmetry.
If we allow negative values of (x,y), then there are actually eight points
associated to
the one known Pythagorean triple: (132, 143), (132, -144), (-133, 143),
(-133, -144),
and the four transpositions. All eight points are also on the circumference
of an
intersecting circle.

Naively we could hope to find another intersection geometry, and iterate
more points,
even if only rationals. But considering the plane section genus 3 and
Faltings's theorem,
the outlook is not hopeful. You seem to agree here.

So many other interesting questions are more nearly in reach. For example,
if we intersect
congruent ellipses:

(1 - x^2 + sqrt(2)*x*y - y^2)=0     and     (1 - x^2 - sqrt(2)*x*y - y^2)=0

The central square-symmetric region has slightly more area than a square of
side
length sqrt(2), A=2.22144 . . . Is this number really A247719, Pi/sqrt(2) ?

Cheers,

Brad


On Thu, Aug 30, 2018 at 2:43 AM Andrew N W Hone <A.N.W.Hone at kent.ac.uk>
wrote:

> Hi Brad and Jon,
>
> The equation
>
> T(x)^2 + T(y)^2 = T(z)^2
>
> is not homogeneous in x,y,z, so it defines an affine surface in three
> dimensions. If there is some way to transform
> it to a quartic plane curve of genus 3, then that would be interesting.
> Please supply details!
>
> Faltings' theorem says that curves of genus g>1 have only finitely many
> rational points. However, getting
> good effective bounds on the size (height) of rational solutions can be
> very hard, and there can be huge solutions.
> This article by Everest and Ward has a nice discussion about this problem:
> https://www.tandfonline.com/doi/abs/10.4169/amer.math.monthly.118.07.584
>  (It is available free on arXiv and Jstor.)
>
> All the best,
> Andy
> ________________________________________
> From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Brad Klee [
> bradklee at gmail.com]
> Sent: 21 August 2018 17:17
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: Pythagorean triples of triangular numbers?
>
> Hi Jon,
>
> Thanks for including a question about geometry! From the
> Diophantine equation,
>
> T(x)^2 + T(y)^2 = T(z)^2,
>
> We can apply a birational change of coordinates and write that,
>
> a = 2*H = x^2 + y^2 - (1/2)*(x^4+y^4) ,
>
> with a new linear parameter "a". H(x,y) is a Hamiltonian family of
> algebraic plane curves, mostly with genus g>1, i.e. not elliptic.
> For curves of higher genus, Falting's theorem says that addition
> rules will not generate an infinite set of points.
>
> However these curves a = 2H(x,y) have interesting symmetry,
> and are really worth more investigation. The singular points are:
>
>     a                       (x,y)                             type
> ============================================
>     0                       (0,0)                           Circular
>    1/2             (0,+/-1),  (+/-1,0)                 Hyperbolic
>     1               +/-(1,1),  +/-(1,-1)                 Circular
>
> And if you plot the function, it's easy to see that the four
> hyperbolic points fall on the intersection of two ellipses,
> and that the circular points fall interior to the ellipses.
>
> Since H(x,y) is an inversion-invariant quartic, we can quickly
> derive the period integral, and then to compute its defining
> differential equation. In less than 1/10 of a second:
>
> 3*(2*a-1)*T(a)+4*(6*a^2-6*a+1)*T'(a)+4*a*(2*a-1)*(a-1)*T''(a)=0,
>
> with integer-scaled solution around a=0 starting:
>
> 1 + 12*a + 228*a^2 + 5040*a^3 + 121380*a^4 + . . .
>  ( Should this integral be included in OEIS? )
>
> The interesting thing about the D.E. for T(a) is that it has only
> three terms, as do the D.E.'s for elliptic curves. As far as I can
> tell, this means that despite g>1, the complex-valued Riemann
> surfaces a=2*H(x,y) are constructed by joining *identical* toric
> sections. If we can figure out this fact, it should lead to some
> really interesting ideas for new, exciting, genus-defying maps.
> This is one of those questions with implications for physics
> as well as number theory.
>
> Cheers,
>
> Brad
>
>
>
> On Tue, Aug 21, 2018 at 3:33 AM <jonscho at hiwaay.net> wrote:
>
> [ Content Trimmed ]
>
> >
>
> > -- elliptic curves offer a smart way to search for solutions?
> >
> > Thanks!
> >
> > -- Jon
> >
> >
>
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