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

Question

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.

Answer 1

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.