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.

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    $\begingroup$ It is definitely an interesting question, but I would say not a research one, so may not be the best suited for this site. That said, the congruent number problem essentially asks for a computational proceedure which 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. $\endgroup$ – Wojowu Nov 9 '19 at 12:13
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    $\begingroup$ For instance if this theorem : en.wikipedia.org/wiki/Tunnell%27s_theorem could be made unconditional, then we would have a procedure. $\endgroup$ – Chris Wuthrich Nov 9 '19 at 12:33
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    $\begingroup$ It's open because no one has solved it yet $\endgroup$ – Stanley Yao Xiao Nov 9 '19 at 14:32
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    $\begingroup$ @StanleyYaoXiao To me the question reads "why doesn't the elliptic curve criterion solve the problem", which basically comes down to what the problem is in the first place (which I hope has now been clarified) $\endgroup$ – Wojowu Nov 9 '19 at 14:43
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    $\begingroup$ For your interest: the book GTM 97 by Koblitz explains all the story, including the dependence on the (weak) BSD conjecture. $\endgroup$ – WhatsUp Nov 9 '19 at 16:19

Pythagorean integer triple coordinates take the general form $$\Big(uv, \frac{v^2 - u^2}{2}, \frac{v^2 + u^2}{2}\Big).$$ This implies that a congruent number $N$ can be expressed in the form $N = uv\frac{v^2 - u^2}{4}$. This second expression implies that there exists some graphical method of determining the rational numbers $u$ and $v$ if $N$ is a congruent number. Fortunately the graph can generate the smallest integers associated to the congruent number and also its multiplication or division with a square number.

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    $\begingroup$ How well does your graph method work for, say, $N=157$? $\endgroup$ – Wojowu Feb 29 at 14:53
  • $\begingroup$ Graphically find the solution of $u = \sqrt[3](\frac{2N}{v}+ \sqrt(\frac{4N}{v^2} - \frac{v^6}{27}))$ - $\sqrt[3](\frac{-2N}{v}+ \sqrt(\frac{4N}{v^2} - \frac{v^6}{27}))$ with the substitution N = 157. $\endgroup$ – Samuel Feb 29 at 15:13
  • $\begingroup$ How do I find it graphically? $\endgroup$ – Wojowu Feb 29 at 17:12
  • $\begingroup$ $u$ and $v$ can be rational, not only integers $\endgroup$ – François Brunault Feb 29 at 18:18
  • $\begingroup$ Feed the function in some geogebra application to obtain the graph and see if you can obtain an integer point.. An integer point like (u, v) = (3, 1) is obtained for example is in case when N is 5, 20, 45, 125 and so forth. $\endgroup$ – Samuel Feb 29 at 18:35

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