How does one find all integer solutions to the equation $3a^2-4b^3=7^c$?
2026-04-26 02:45:04.1777171504
On the equation $3a^2-4b^3=7^c$
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I have a feeling that the solutions I gave are the only ones, but I don't have a proof of that. (Those solutions are the families $(\pm6^{3m},-7^{2m},1+6m)$ and $(\pm13\cdot6^{3m},-5\cdot7^{2m},1+6m)$ for $m\in\mathbb{N}$.)
Maybe somebody who knows more about elliptic curves can pick this up. For any value of $c$, you can look at the elliptic curve $y^2=x^3+2^4\,3^3\,7^c$. For fixed values of $c$ it seems you can show that the torsion part is trivial and that the curves are of rank 0 unless $c\equiv1\pmod{3}$. For $c\equiv1\pmod{6}$ you'll get the two solutions given above coming from powers of the generator with the rest of the powers giving non-integral rational solutions. (For $c\equiv4\pmod{6}$ points on the curve do not correspond to integral solutions, since one can easily check the requirements $a\equiv1\pmod2$, $b\equiv2\pmod3$ and $c\equiv1\pmod2$ for integer solutions to the original equation.)
I don't have a general argument for these facts (just checked numerous special cases) so this is just hand-waving for now, but I'm not a number-theorist, so I'll let the experts fill in the details...