Re: question on solutions of rational versus integer ternary quadratics

[William Jagy <wcjagy@gmail.com>]

I sent you an email a few hours ago, pointing out that you had introduced a mistaken coefficient (1779 rather than the correct 1739) , at least in the top line of one of your output files

On Mon, Jul 14, 2025 at 1:12 PM American Citizen <website.reader3@gmail.com> wrote:

Hello:

I am working with the following conic and 2 ternary quadratics

conic =  [261/5, -3481/50, 261/5, -1739/100, -1739/100, -1]
T9Q([x,y,z]) = 261/5*x^2 + (-3481/50*y - 1739/100*z)*x + (261/5*y^2 -
1739/100*z*y - z^2)
T9Z([x,y,z]) = 5220*x^2 + (-6962*y - 1739*z)*x + (5220*y^2 - 1739*z*y -
100*z^2)

Over the range of integers x,y,z, for -10,000 <= x,y,z <= 10,000 I found
2367 solutions for T9Q, but only 1076 for T9Z. However the 1291
solutions from T9Q plugged into T9Z do work.

Why is T9Q missing 1,291 solutions?

Randall

btw: These ternary quadratic solutions collapse to just 113 points on
the conic, but the [0,0] point has to be removed.

Another btw note: I had special rational points on conics, it takes 2
rational points to successfully recover all rationals on the conic,
using the point-slope method to parameterize, and the reason is simple,
for y - y1 = m(x-x1) where [x1,y1] is known and m is the slope, if you
set m=0 for slope zero, the equation collapses to y - y1 = 0, and that
only recovers your original [x1,y1] point and misses the 2nd y point
which IS on the conic. I had to use 2 points and only found this out
after laboring on this for at least 3 days wondering why my rational
lattice points were missing some rationals.