In his book Elliptic Curves, A. Knapp illustrates the close relationship between the proof of Mordell's Theorem and Fermat's proof (both via infinite descent) that the equation $u^4 + v^4 = w^2$ has no nontrivial integer solutions.
To do so, Knapp first shows how to associate an elliptic curve to the given equation. In fact, after dividing by $v^4$, we are interested in the rational solutions to the equation $$y^2= x^4 + 1$$
Of course, this is still no Weierstraß Form. But we can make the following non-projective (albeit bijective and birational) change of variables $x\mapsto x$ and $y\mapsto y+x^2$ in order to obtain the following homogenized equation $$Y^2W + 2X^2Y = W^3$$ It's only $\mathbb{Q}$-rational flex is at $(1:0:0)$, where the tangent line is $y=0$. So after two linear transformations mapping $(1:0:0)$ to $(0:1:0)$ and letting the tangent become $w=0$, and after scaling properly we are left with the affine equation $$ y^2= x^3 - 4x \tag{$\dagger$} \label{eq:ellipcuv} $$
Notice that since the original equation only had trivial solutions, the rational solutions of $(\ref{eq:ellipcuv})$ are $\infty, (0,0)$ and $(\pm 2, 0)$.
Later in the book, when proving Mordell's Theorem that the group of rational points on any elliptic curve is finitely generated, Knapp begins by examining Fermat's proof via infinite descent that $u^4+v^4 = w^2$ has no nontrivial solutions, and notes how the steps Fermat follows reveal information about the associated elliptic curve $(\ref{eq:ellipcuv})$:
Thus Fermat's method of descent amounts to a construction that starts with $P'$ in $E(\mathbb{Q})$ and constructs $\pm \frac12 P'$ in $E(\mathbb{Q})$. Closer inspection shows that we made certain adjustments to our variables as we went along, in order to be able to carry out the descent. The first one was to assume that $u$ is odd and $v$ is even, rather than the other way around. This is the first adjustment. The next assumption was that $u$ and $v$ are positive; if $u$ and $v$ are not positive, we adjusted them to make them positive. We chose $r$ and $s$ to be the positive square roots of $p$ and $q$, but we could not force $r$ to be odd and $s$ to be even. Normalization of $r$ and $s$ to be of the same form as $u$ and $v$ was a second instance of the first adjustment. In the context of the elliptic curve, it turns out that both adjustments amount to changing the point we are studying by a coset representative of $E(\mathbb{Q})/2E(\mathbb{Q})$. The fact that at most two adjustments are needed reflects the fact that $E(\mathbb{Q})/2E(\mathbb{Q})$ is a sum of only two copies of $\mathbb{Z}_2$ in this example. [p. $83$]
The question. Although I doubt very much that, historically, this was some kind of motivation for posing the admittedly beautiful but also quite opaque proof of Mordell's Theorem, I wondered if this just happens to be a mere coincidence, or if there is any known larger family of diophantine equations for which the proof of the non-existence of non-trivial solutions reflects on their associated elliptic curve, for instance, the fact that we actually want to focus on $E(\mathbb{Q})/2E(\mathbb{Q})$ and that it would be great if the index $(E(\mathbb{Q}):2E(\mathbb{Q}))$ was finite.