I have already tried a somewhat exhaustive search of the literature, but couldn't find anything close to the problem that I am working on.
My question is: When does Mordell's Equation
$$y^2 = x^3 + K$$
have only FINITELY many solutions over the field of rational numbers, if we allow $K$ itself to be a rational number?
I've seen a "criterion" (i.e. a set of sufficient conditions) related to the class number of the (real/imaginary) quadratic field $\mathbb{Q}(\sqrt{K})$, but it is limited only to $K$ being either 1 or 2 modulo 4.
[The actual "criterion" (as stated in the Japanese[?] paper that I allude to) is:
Mordell's equation $y^2 = x^3 + K$ has finitely many solutions in $\mathbb{Q}$ if
(1) $-K$ is not of the form $3t^2 + 1$ or $3t^2 – 1$; AND
(2) $K \equiv 1 (mod 4)$ or $K \equiv 2 (mod 4)$; AND
(3) $3$ does not divide the class number of the (real/imaginary) quadratic field $Q(\sqrt{K})$.]
Edit: Please refer to this hyperlink for more information as to the context of the previous "criterion". These have since been refuted by Kevin Buzzard (@Kevin – thank YOU!).
Thanks to Kevin for pointing out some of the subtle errors in my original post, particularly in the third condition. I was considering the case $K > 0$ (i.e. for real quadratic fields).
Now for the context:
Let
$$Y = W + Z$$
and
$$X = WZ$$
where $W$ and $Z$ are defined as:
$$W = I(p^k) = \frac{\sigma_{1}(p^k)}{p^k}$$
$$Z = I(m^2) = \frac{\sigma_{1}(m^2)}{m^2}$$
Let $$N = {p^k}{m^2}$$ be a perfect number. (At this point, we don't have to distinguish between even or odd $N$ because the Euclid-Euler model for perfect numbers fits both cases. For more details regarding this, please refer to this link.)
We "know" that the exponent $k$ allows us to distinguish between even and odd $N$ in the sense that:
(1) If $k$ = 1, then $N$ is even.
(2) If $k$ > 1, then $N$ is odd. (Again, refer to the link for more details. There is also a related MathOverflow post here.)
Thus, a (possibly) feasible and modern approach to the OPN problem (i.e. determining nonexistence or otherwise) will be to try establishing a finiteness result first (for particular values of $K$).
In other words, checking for finiteness of OPNs amounts to checking for finiteness of solutions for Mordell's equation
$$Y^2 = X^3 + K$$
for particular values of $K$.
And you will only have to check for values of $K$ in the range $[50, 399]$ (for a total of 350 elliptic curves), per the previous answer to this MathOverflow question.
$K$ falls in that range because the sum
$$Y = W + Z$$
is known to lie in the open interval $(57/20, 3)$.
Of course, the "juicy" implication is that: If you will be able to find a condition (e.g. equation, inequality, etc.) relating $k$ to $K$ and you are also able to FURTHER show that the number of solutions to the corresponding Mordell equation $Y^2 = X^3 + K$ is finite FOR ALL SUCH $K$, then it would follow that there are only finitely many perfect numbers (odd AND even).
Disclaimer: This is a "naive" approach based on my current understanding of elliptic curve theory. I am well-aware, of course, that the rationals are dense over the real numbers. [Edit: In addition, the abundancy indices and the abundancy outlaws are both dense over the rationals.] Which is why I was kinda surprised that there is NO need to assume ("strict") rationality (i.e. $K \in \mathbb{Q}$ but not in $\mathbb{Z}$) for $K$ when checking for finiteness of solutions to Mordell's equation.
Best Answer
First of all, if you replace $k$ by $d^6k$ you get another equation such that the corresponding sets of rational solutions are in bijection. So, you might as well assume that $k$ is an integer. I don't think there is a simple, crisp criterion for the equation to have finitely many solutions. Birch-Swinnerton-Dyer predicts that this is the case if and only if the $L$-function of the elliptic curve does not vanish at $s=1$ and the "if" part is known (Coates-Wiles). There is no shortage of literature on that.