Number of integral solutions to $4y^{3}=3x^{2}+1$ are?

Please help with this, i formed an equation for $y$ but don't know how to find the solutions or the number of solutions.

  • $\begingroup$ computer check for $1 \leq x,y \leq 1000$ give the only solution $(1,1)$ and also $(-1,1)$ is valid. $\endgroup$ – Ahmad May 28 '17 at 10:14
  • $\begingroup$ It barely is an elliptic curve. $\endgroup$ – enedil May 28 '17 at 23:06

Looking at the equation modulo $2$ we see that $x \equiv 1 \pmod{2}$, so let's put $x=2k+1$. After simplification the equation then becomes

$$ y^3=3k^2+3k+1. $$ Now if we add $k^3$ to both sides, we end up with

$$ y^3+k^3=k^3+3k^2+3k+1=(k+1)^3. $$

Perhaps you recognize the $a^3+b^3=c^3$ equation in it... By Fermat's Last Theorem there are no solutions except trivial ones where one of the numbers is $0$. Now just inspect those cases and you find all solutions.

The proof of FLT for $n=3$ can be done quite elementary, so you might want check that as well to get more understanding of that part of the proof.

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