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.
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.
If you can reduce the equation to where is square-free, then it might work. The first step is to note . And so . And we are left with an equation, , but here are not integers. Perhaps you can take advantage of the fact that has unique factorization and use it to prove that has not only solutions in but more, no solutions in .