<s>[INST] Ramanujan, sum of two cubes - how was it discovered? I'm looking for motivation for, and hopefully a derivation of, Ramanujan's sum of  cubes formula 
$$\left(x^2+7xy-9y^2\right)^3+\left(2x^2-4xy+12y^2\right)^3=\left(2x^2+10y^2\right)^3+\left(x^2-9xy-y^2\right)^3$$
I can't see where one could start to justify this without actually expanding everything out. 
Additionally, does it provide a parametrisation of the surface 
$$X^3+Y^3=Z^3+1$$ in $\mathbb{Q}^3$
with 
$$X=\frac{x^2+7xy-9y^2}{2x^2+10y^2}$$
$$Y=\frac{2x^2-4xy+12y^2}{2x^2+10y^2}$$
$$Z=\frac{x^2-9xy-y^2}{2x^2+10y^2}$$
 [/INST] "OP" inquired about finding parameterization of two cubes in two different way's.
Consider the below equation;
$s^3+t^3=u^3+v^3$  ----$(1)$
Tomita has given a method to parameterize equation $(1)$. 
In his solution, take $(p,q,r,s)=(-1,9,10,12)$ 
then the below mentioned solution is arrived at:
$s=36x^2+22xy+y^2$
$t=40x^2-40xy+12y^2$
$u=48x^2-40xy+10y^2$
$v= 4x^2+22xy+9y^2$
For, $(x,y)=(0,1)$, we get, 
$(s,t,u,v)=(1,12,10,9)$
His webpage link is given Here
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