Some **experimental results** for the **Markoff-Hurwitz equation** :
$$X_{1}^{2} + X_{2}^{2} + ... + X_{r}^{2} = mX_{1}X_{2}...X_{r} $$ for $m \ge 1$, $X_{i} \in \mathbb{N}$ and $1 \le X_{1} \le X_{2} \le ... \le X_{r} $

Let $n = mX_{1}X_{2}...X_{r}$.

**Remark** : If $(a_{1},a_{2}, ... ,a_{r})$ is a solution, then $(a_{1},a_{2}, ... ,a_{r-1},b_{r} )$ also (up to sorting), with $b_{r} =ma_{1}a_{2}...a_{r-1} - a_{r} $.
It follows that each solution comes from a **fundamental**, checking : $$mX_{1}X_{2}...X_{r-1} \ge 2 X_{r}$$ Here are the *fundamental solution* computed for $n<100000$ and $3 \le r \le 14$:

**rank $r=3$ :**

$m=1$ : $(3,3,3)$ : $n=27$

$m=3$ : $(1,1,1)$ : $n=3$

**rank 4 :**

$1$ : $(2,2,2,2)$ : $16$

$4$ : $(1,1,1,1)$ : $4$

**rank 5 :**

$1$ : $(1,1,3,3,4)$ : $36$

$4$ : $(1,1,1,1,2)$ : $8$

$5$ : $(1,1,1,1,1)$ : $5$

**rank 6 :**

$3$ : $(1,1,1,1,2,2)$ : $12$

$6$ : $(1,1,1,1,1,1)$ : $6$

**rank 7 :**

$1$ : $(1, 1, 1, 2, 2, 2, 3)$ : $24$

$2$ : $(1, 1, 1, 1, 2, 2, 2)$ : $16$

$3$ : $(1, 1, 1, 1, 1, 2, 3)$ : $18$

$5$ : $(1, 1, 1, 1, 1, 1, 2)$ : $10$

$7$ : $(1, 1, 1, 1, 1, 1, 1)$ : $7$

**rank 8 :**

$1$ : $(1, 1, 1, 1, 2, 2, 2, 4)$ : $32$

$8$ : $(1, 1, 1, 1, 1, 1, 1, 1)$ : $8$

**rank 9 :**

$6$ : $(1, 1, 1, 1, 1, 1, 1, 1, 2)$ : $12$

$9$ : $(1, 1, 1, 1, 1, 1, 1, 1, 1)$ : $9$

**rank 10 :**

$1$ : $(1, 1, 1, 1, 1, 1, 1, 3, 4, 4)$ : $48$

$2$ : $(1, 1, 1, 1, 1, 1, 1, 2, 2, 3)$ : $24$

$4$ : $(1, 1, 1, 1, 1, 1, 1, 1, 2, 2)$ : $16$

$6$ : $(1, 1, 1, 1, 1, 1, 1, 1, 1, 3)$ : $18$

$10$ : $(1, 1, 1, 1, 1, 1, 1, 1, 1, 1)$ : $10$

**rank 11 :**

$2$ : $(1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 4)$ : $32$

$3$ : $(1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3)$ : $27$

$7$ : $(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2)$ : $14$

$11$ : $(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)$ : $11$

**rank 12 :**

$12$ : $(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)$ : $12$

**rank 13 :**

$1$ : $(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 5)$ : $60$

$3$ : $(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4)$ : $36$

$4$ : $(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3)$ : $24$

$7$ : $(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3)$ : $21$

$8$ : $(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2)$ : $16$

$13$ :$(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)$ : $13$

**rank 14 :**

$1$ : $(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 6)$ : $72$

$1$ : $(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3)$ : $36$

$4$ : $(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4)$ : $32$

$5$ : $(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2)$ : $20$

$14$ : $(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)$ : $14$

**Remark** : At rank $14$ and $m=1$ there are two fundamental solutions.

**Observations** on the fundamental solutions (most of them are already known and proved) :

- This is a classification for $3 \le r \le 14$.
- $X_{1} \ne 1$ implies $X_{1} = ... = X_{r} \in \{ 2,3 \}$ and $r \in \{ 3,4\}$.
- Finiteness at fixed $r$.
- $1 \le m \le r$
- $X_{r}< 2r/m$ (open ?)
- Will Jagy's conjecture : $X_{r} \le \sqrt{9(r+6)/5}$

**Graphics**: for $3 \le r \le 2500$ (5 days of computer calculations)

**1.** The maximum $M(r)$ for $n$ at a fixed $r$ (for the fundamental solutions):

*A repeating pattern is observed.*

The stratification comes probably from the minimum value of $m$ for $r$ fixed.

**2.** The number $N(r)$ of fundamental solutions at a fixed $r$ :

**Conjecture on upper-bounds** (open) :

- $M(r) \le 3.6r+21.6 $
- $N(r) \le ln(r)^{2+ln(ln(ln(r)))/2}+2 $