Two years ago, I made a conjecture [on stackexchange](https://math.stackexchange.com/questions/334448/1-a21-b21-c2-8abc-a-b-c-in-mathbbq-has-infinitely-many-sol/336803#336803):


Today, I tried to find all solutions in integers $a,b,c$ to
$$(1-a^2)(1-b^2)(1-c^2)=8abc,\quad a,b,c\in \mathbb{Q}^{+}.$$ 


I have found some solutions, such as
$$(a,b,c)=(5/17,1/11,8/9),(1/7,5/16,9/11),(3/4,11/21,1/10),\cdots$$


$$(a,b,c)=\left(\dfrac{4p}{p^2+1},\dfrac{p^2-3}{3p^2-1},\dfrac{(p+1)(p^2-4p+1)}{(p-1)(p^2+4p+1)}\right),\quad\text{for $p>2+\sqrt{3}$ and $p\in\mathbb {Q}^{+}$}.$$


Here is another simple solution:
$$(a,b,c)=\left(\dfrac{p^2-4p+1}{p^2+4p+1},\dfrac{p^2+1}{2p^2-2},\dfrac{3p^2-1}{p^3-3p}\right).$$



> 
> My question is: are there solutions of another form (or have we found all solutions)?
> 
> 
> 



Find all solution $a,b,c$ with $(1-a^2)(1-b^2)(1-c^2)=8abc$
Call (the projective completion of) your surface $S$. It admits three double covers of $\mathbb P^2$, namely $$\pi\_1(a,b,c)=(a,b), \quad\pi\_2(a,b,c)=(a,c),\quad\pi\_3(a,b,c)=(b,c).$$ Each double cover induces an involution, so we get
three involutions $$\sigma\_1,\sigma\_2,\sigma\_3:S\to S.$$ These involutations don't commute, and if you form $f=\sigma\_1\circ\sigma\_2$ and $g=\sigma\_1\circ\sigma\_3$, then $f$ and $g$ generate a subgroup $G$ of $\text{Aut}(S)$ that is a free group on two generators. And presumably for most starting points $P\in S(\mathbb{Q})$, repeated application of $f$ and $g$ will give you a tree of rational solutions. (You might compare this with the generation of all positive integer solutions to the Hurwitz equation $x^2+y^2+z^2=3xyz$ starting from $(1,1,1)$.) Here are some references to papers that have studied rational points on K3 surfaces admitting 3 involutions:


* Baragar, A. Rational points on K3 surfaces in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$. *Math. Ann.*
**305** (1996), no. 3, 541–558.
* Wang, L., Rational Points and Canonical Heights on K3-surfaces in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$, *Contemporary Math.* **186** (1995), 273 – 289


