a question I answered,
has apparently not helped the student enough. I answered his email today, maybe that will help. In particular, I wrote out the Cayley-Hamilton thing with no matrices.
Meanwhile, this exact diophantine equation has been asked about before on Main. I am unable to locate such, partly because there is such variety in the way people ask about these things, many disguises.
So that is the question, what are a few previous instances of, essentially, this diophantine equation on Main?
EDIT: I did try google with
x^2 - 3 x = 2 y^2 site:math.stackexchange