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 $