Name of Formula $x^3+y^3=z^3+1$ I encountered the formula
$$x^3+y^3=z^3+1$$
with the condition, that
$$x<y<z$$
and wonder, whether it has got a specific name or whether it can be easily transformed into another well-known (family of) formula(s).

$$X^3+ Y^3+ Z^3=1$$ is the formula which is known as harder factor and yours is a distorted and conditional form of harder factor
If $X+Y+Z=0$ then $X^3+ Y^3+ Z^3=1$.
In your question $X$ is less than $Y$ and $Y$ is less than $Z$ means the minimum possible difference between $X$ and $Y$, $Y$ and $Z$ is $1$.
At the same time the minimum possible difference between $X$ and $Z$ will be $2$. 
So there will in all the cases except $X=-2$, $Y=-1$, $Z=3$ where $X+Y+Z$ is not equal to zero then it must be that $X^3+ Y^3+ Z^3$ is not equal to $1$. So $X^3+ Y^3+ Z^3$ must be greater/less than $1$. As it is given that $Z>Y>X$ then $X^3+ Y^3$ must be unequal to $Z^3$. It means 
$X^3+ Y^3$ may be equal to $Z^3+1$. 
In this way $X^3+ Y^3+ Z^3=1$ is related to the question asked by the poster
