# Different definitions of Lucas groups

I see two different definitions of Lucas groups, stated below.

Is one of the two standard? Are they trivial variations?

From these slides (Liljana Babinkostova et al., Boise State University, 2017)

The Lucas roup for parameters $$(D,N)$$ is the set $$\bigl\{(x,y)\in{\Bbb Z_N}^2,\ x^2-D\,y^2\equiv1\pmod N\bigr\}$$ with the group law defined by $$(x_0,y_0)\cdot(x_1,y_1)=\\ \bigl((x_0\,x_1+D\,y_0\,y_1)\bmod N,\ (x_0\,y_1+y_0\,x_1)\bmod N\bigr)$$

From that archived online course (same source, 2016)

It is restricted to $$D\ge5$$ with $$D+4$$ a square, and $$N\ge3$$ coprime with $$2D$$.

The Lucas roup for parameters $$(D,N)$$ is the set $$\bigl\{(x,y)\in{\Bbb Z_N}^2,\ x^2+D\,y^2\equiv4\pmod N\bigr\}$$ with the group law defined by $$(x_0,y_0)*(x_1,y_1)=\\ \biggl(\frac{N+1}2(x_0\,x_1+D\,y_0\,y_1)\bmod N,\ \frac{N+1}2(x_0\,y_1+y_0\,x_1\bmod N\biggr)$$