The Mordell equation is the Diophantine equation $$Y^2 = X^3-k \tag{1}$$ where $k$ is a given integer. There is no known single method — elementary or otherwise — to solve equation $(1)$ for all $k$, though there are various ad hoc methods which work on large classes of $k$, and there are [published] elementary solutions for many (though not all) $k$.
For the scope of this discussion, I define “elementary” as pre-college level number theory — definitely excluding imaginary numbers, class numbers, imaginary quadratic fields, "advanced" UFDs, and so on; almost certainly excluding quadratic reciprocity and similar; but definitely including FTA (unique factorization in the regular integers), mathematical induction, polynomial division, simple modular arithmetic, and so on.
Emboldened by my recent completely elementary solution to the Mordell equation for $k=-1$, I am wondering if an elementary proof can be found for every Mordell equation. Since it is impossible to prove a negative, I guess my real question is this:
For any integer or class of integers $k$, are there any obvious impediments to the development of an elementary solution?
There is a well known elementary proof that the equation
$$ y^2 = x^3+7 $$
has no solution in integers $x,y$.
If $x$ is even, then $y^2 \equiv 3\pmod{4}$. Since this is not possible, $x$ must be odd. Hence $x^2-2x+4 \equiv 1+2x \equiv 3\pmod{4}$, and so there exists a prime $q$ such that $q \mid (x^2-2x+4)$ and $q \equiv 3\pmod{4}$. But then
$$ y^2+1 = x^3+8 = (x+2)(x^2-2x+4) $$
implies $q \mid (y^2+1)$, which is impossible for primes of the form $4k+3$.
We have shown that the Mordell equation corresponding to $k=7$ has no integer solutions. $\blacksquare$