1

Let $Q$ be the ternary quadratic form $Q(x,y,z)=x^2+y^2-z^2$. For $k\in{\mathbb Z}$, denote by $E_k$ the equation $Q(x,y,z)=k$, to be solved in integers $x,y,z$.

It is easy to see that $E_k$ always has a solution (when $k$ is odd, $k=2t+1$, we have $k=Q(t+1,0,t)$, and when $k$ is even, $k=2t$, we have $k=Q(3t,4t-1,5t-1)$).

Is there a parametrization known giving all the solutions of $E_k$ ? Or parametrization with many variables, giving many solutions ?

Ewan Delanoy
  • 61,600
  • http://math.stackexchange.com/questions/575931/what-integers-can-be-represented-by-the-quadratic-form-4x2-3y2-z2/777067#777067 – individ Aug 14 '16 at 08:22
  • http://math.stackexchange.com/questions/165698/on-the-diophantine-equation-a2b2-c2k – mercio Aug 14 '16 at 08:50

1 Answers1

0

The equation can be represented as a difference of squares.

$$x^2+y^2-z^2=k$$

Then asking a number you can write down the solution, presenting a $t$ as one of the multipliers.

$$y=y$$

$$x=\frac{y^2-k}{2t}-\frac{t}{2}$$

$$z=\frac{y^2-k}{2t}+\frac{t}{2}$$

individ
  • 4,301