I was reading up about the congruent number problem.

One of the theorems on the subject says how the two things are equivalent: a positive integer $n$ being a congruent number and elliptic curve $y^2 = x^3-n^2 x$ having a non trivial rational solution. Later on, the notes say that the Congruent Number Problem is still an open one.

Now my question is, doesn't the above result give a criterion to determine whether a given positive integer is a congruent number or not? Could the reason be that we don't have a way of knowing when such elliptic curves have a non trivial solution and hence no straight way of knowing if $n$ is a congruent number? And wait, when we say, we're looking for a criterion, what exactly do we mean?

P.S.: I'm new to the theory of elliptic curves so apologies if the above question seems a bit ignorant or not per MO's standards.

computational proceedurewhich will tell us, in finite time, whether $n$ is a congruent number. As you realize, this reduces to the problem of determining whether an elliptic curve has a rational point, but we don't have a (provably correct!) methods to verify that. $\endgroup$1more comment