,
such as y2=x3-x,
there are only finitely many rational points,
while for others, such as the example
y2=x(x+5)(x-5) above,
there are infinitely many.
It turns out that for some elliptic curves over
,
such as y2=x3-x,
there are only finitely many rational points,
while for others, such as the example
y2=x(x+5)(x-5) above,
there are infinitely many.
But in any case there is a deep theorem, proved by Mordell,
that says that the group of rational points on an elliptic curve Eis ``finitely generated.''
This means there is a finite list of rational points
on Esuch that all rational points on E can be generated
from the points in
by iteratively applying +to pairs of points.
On the other hand, it is not known whether there exists an algorithm
that takes the equation of an elliptic curve
and outputs a finite list
of generating points as above.
Researchers in number theory have spent about 70 years trying to prove
the existence of such an algorithm,
but the problem is still unsolved!