Given diophantine equation $4x^3 - 3 = y^2$ ($x > 0$). How many solutions are there ?
I don't know where to start, please give me a hint
This is an elliptic curve, with general Weiertsrass equation $$ y^2+y=x^3-1. $$ To see this, just use the substitution $(x,y)\mapsto (x,2y+1)$ for $y^2=4x^3-3$. Its integral points are given by $(1,0), (1,-1),$ and $(7,18), (7,-19)$, hence the integer points of $y^2=4x^3-3$ are given by $$ (1,1),(1,-1),(7,37),(7,-37). $$ The rank is $1$, i.e., there are infinitely many rational points on the curve.