I want to find the integer solutions to this Diophantine equation: $$5x^3=y^2+1$$ I have seen a lot of problems with monic variables, but not with a constant on the $x^3$ such as this.
I know I can factorise the right hand side and get $5x^3=(y-i)(y+i)$, so I can work in $\mathbb Z[i]$. But I am unsure where to proceed from here, and how the $5$ comes into the problem.
If we multiply both sides by $25$ and rearrange we get $$(5y)^2=(5x)^3-25$$ So an integral solution to the original solution corresponds to an integral point on the elliptic curve $w^2=z^3-25$. But according to the LMFDB there are only two integral points on the curve, $(z,w)=(5,\pm10)$. We conclude that the only integral solutions to the original equation are $(x,y)=(1,\pm2)$.
Letting $z=Nx,w=Ny$ shows that integral solutions to $Nx^3=y^2+1$ for fixed $N$ correspond to integral points on the Mordell curve $w^2=z^3-N^2$, so there are always finitely many solutions and they may be rather effectively counted. In the $N=17$ case, the only solutions are $(x,y)=(1,\pm4)$.