Regarding solvability of an Elliptic Curve over Q

Question

Prove that infinitely many solutions may be found over $\mathbb{Q}$ for the elliptic curve : $y^2=x^3-9x+9$

Answer 1

The theory of elliptic curves has progressed a lot and this can be proved easily by the idea of group law on elliptic curve (It was introduced by Bachet centuries ago).

An elliptic curve over rational numbers is a curve given by $y^2=x^3+Ax+B$ such that $A,B \in \mathbb{Q}$ and discriminant of the RHS polynomial is nonzero( so that the curve be nonsingular).

You can sum two rational points of an elliptic curve in a natural way and this method will give you a group law on the rational points and "the infinity point" of elliptic curve.

So now suppose we found a rational point $P$, then you can consider $2P= P+P$ and get "probably" another rational point and then consider $3P=P+P+P$ and again get "probably" another rational point and so on. So if we knew that we never get back to old points, we could find infinitely many rational points on the curve.

There is a theorem called Nagell-Lutz theorem which says that if the elliptic curve is given by integer coefficients (like yours), and if $P$ is a rational point on the curve, If for some distinct $n,m \in \mathbb{N}$ we had $nP=mP$ then $P $ has integral coordinates.

So if we can find a rational point on a curve which is not integral, we can consider $ \lbrace nP : n \in \mathbb{N} \rbrace$ and by the theorem this set would give us an infinite number of rational points.

So, in your example we can consider $(x,y)=(\frac{9}{4},\frac{3}{8})$ and we are done ;)