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I started reading Silverman and Tate’s introductory book on elliptic curves. In the introductory chapter they mention that for the Bachet equation $x^2 - y^3 = c$, there are infinitely many rational solutions for most $c$, but finitely many integer solutions.

Do we know anything about solutions $(x,y) \in \mathbb{Q} \times \mathbb{Z}$? How about solutions of the form $x = \frac{p}q$ for primes $p,q$ (or more generally almost primes) and $y \in \mathbb{Q}$? Are there catalogues of such easily stated questions and their (partial) progresses somewhere?

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    $\begingroup$ Just a quick remark: If $c \in \mathbb{Z}$, which I believe was the original context, then every rational solution with $y \in \mathbb{Z}$ must also have $x \in \mathbb{Z}$. (The square root of an integer is rational if and only if it's an integer.) Also -- again, assuming $c \in \mathbb{Z}$ -- there can be no rational solution with $x = p/q$ just by considering $q$-adic valuations. (The denominator of $x^2 - c$ would be $q^2$, but the power of $q$ appearing in the denominator of $y^3$ must be divisible by $3$. $\endgroup$
    – John Doyle
    Commented Dec 31, 2022 at 11:44
  • $\begingroup$ @JohnDoyle ah yes. I realized that also. Hopefully the case of rational c is still worth asking $\endgroup$
    – John Jiang
    Commented Dec 31, 2022 at 13:54
  • $\begingroup$ I guess both questions are equivalent to asking for integer solutions of $a x^2 + b y^3 = c$, which is well studied. $\endgroup$
    – John Jiang
    Commented Dec 31, 2022 at 14:12

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