# Find all solutions of this diophantine equation of the second degree in three variables

Consider the Diophantine equation $Q(x,y,z)=1$, where $Q(x,y,z)$ is the quadratic form $x^2+y^2-z^2$. Let $S \subseteq {\mathbb Z}^3$ denote the set of all solutions. It is rather easy to find several parametric solutions, but it seems harder to find a complete enumeration of all the solutions. Specifically, say that a subset $T$ of ${\mathbb Z}^3$ is a polynomial image when there is a $r>0$ and three polynomial maps $P,Q,R : {\mathbb Z}^r \to {\mathbb Z}^3$ such that $T$ is the image of $(P,Q,R)$. My question is : Can $S$ be written as a finite union of polynomial images?

The older stackexchange question Integral solutions of $x^2+y^2+1=z^2$ is about the same equation, but it does not answer the specific question above.

-

## 1 Answer

Let $\Gamma$ the set of orthogonal unimodular matrices, that is $$\Gamma=M(n,\mathbb Z) \cap O(n).$$ This group acts by left multiplication on the set $S$ of solution of your equation. Moreover, the set of $\Gamma$-orbits in $S$ is finite! That is, there is a finite set $F$ in $S$ such that: for any $X\in S$ there are $g\in\Gamma$ and $Y\in F$ such that $$X=gY.$$ I think this allows you to write $S$ as a finite union of polynomial images. Am I right?

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 22, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9535337090492249, "perplexity_flag": "head"}