Please help me to find a source/reference for solving the equation (or similar) 4u^3 - v^2 = 3.
I am guessing that the only solutions are (1,1), (1,-1), (7,37), and (7,-37), but my efforts have not been successful.
Thanks,
Wayne
Please help me to find a source/reference for solving the equation (or similar) 4u^3 - v^2 = 3.
I am guessing that the only solutions are (1,1), (1,-1), (7,37), and (7,-37), but my efforts have not been successful.
Thanks,
Wayne
Yes. This looks like a problem with Elliptic curves (hence my avatar). A book on Elliptic curves might mention something about this equation. I have a book at home higher number theory, it mentions some stuff, but nothing like this.
Another idea:
If you can reduce the equation to $\displaystyle y^2 = x^3 - k$ where $\displaystyle k$ is square-free, then it might work. The first step is to note $\displaystyle (2u)^3 = 2v^2 - k$. And so $\displaystyle (2u)^3 = (\sqrt{2} v)^2 + 6$. And we are left with an equation, $\displaystyle x^3 = y^2 + 6$, but here $\displaystyle x,y$ are not integers. Perhaps you can take advantage of the fact that $\displaystyle \mathbb{Z}[\sqrt{-2}]$ has unique factorization and use it to prove that $\displaystyle x^3 = y^2 + 6$ has not only solutions in $\displaystyle \mathbb{Z}$ but more, no solutions in $\displaystyle \mathbb{Z}[\sqrt{-2}]$.