I. Elliptic curves

Given integers $a,b,m_k$. Let,

$$x^2+a = m_1u_1^2\\x^2+b = m_1u_2^2\tag1$$

If there is a rational point $x_i$, then the pair (after a transformation) is birationally equivalent to an *elliptic curve*, call it $E_1$, and frequently has infinitely many rational points. Assume a second $m_2$,

$$x^2+a = m_2v_1^2\\x^2+b = m_2v_2^2\tag2$$

with a different rational point $x_j$, and yielding an elliptic curve $E_2$.

II. Question

**Q1:** What is the relationship between $m_1,m_2$ given that ALL the rational points on $(1),(2)$ are on the single elliptic curve $E_3$,

$$(x^2+a)(x^2+b) = y^2\tag3$$

Or knowing the generators of $E_3$, *can we predict what and how many square-free integer $m_k>1$ are permissible?*

III. Example

For simplicity, assume the special case of $b = -a$ which makes $m_1=1$ as the *congruent number problem*. Let $a=101$. For,

$$(x^2+a)(x^2-a) = y^2\tag4$$

two solutions are $x_1 = \frac{2015242462949760001961}{118171431852779451900}$, and $x_2 =\frac{2125141}{63050}$. As applied to,

$$x^2+101 = m_k u_1^2\\x^2-101 = m_k u_2^2\tag5$$

the point $x_1$ also solves $m_1 = 1$, while $x_2$ also solves $m_2 = 101$. *(What other $m_k$ is permissible?)*

**Q2:** In general, if $(5)$ is rationally solvable for one integer $m_k$, does it imply finitely (or infinitely) many other square-free integer $m_k > 1$?