Almost: you have to assume that p, q are congruent to 1 mod 3, not only
modulo the square root.
As for the original questions by Jim Propp:
> 1) Bachet considered some equations of this form (or perhaps the class
> of all of them), thus resulting in their being called the "Bachet
> equation"; I do not know whether he considered this case in particular.
Bachet showed how to compute new rational solutions to y^2 = x^3 - c
from known ones (by the method of tangents, as we call it today,
although apparently without being aware of the geometrical
interpretation).
> 3) Euler gave a proof using complex numbers, but he didn't justify all
> his steps; in particular, he implicitly assumed unique factorization
> for expressions involving the square root of -3.
Correct if you replace -3 by -2: in fact,
y^2 + 2 = (y+\sqrt{-2})(y-+\sqrt{-2}).
> 4) Gauss proved unique factorization for the Gaussian integers (in
> his successful pursuit of a law of quartic reciprocity); he indicated
> that one would need to do something similar with what are now called
> the Eisenstein integers if one wanted a law of cubic reciprocity, but
> did not actually do this.
Though he did not publish on cubic reciprocity, Gauss proved that
x^3 + y^3 = z^3 only has the trivial solutions in Eisenstein integers;
for this he needed unique factorizations. This is in his collected
works: {\em Zur Theorie der komplexen Zahlen}, Werke II (1876),
387--398.
> 6) Kummer did lots of work on cyclotomic fields, but he was stymied
> by quadratic fields in which the discriminant is 1 mod 4, because
> he didn't use the right definition of "algebraic integer".
Kummer did not have any definition of an algebraic integer at all.
He did his computations in fixed orders, which in the cyclotomic
case just happened to coincide with the ring of integers. He
also considered Kummer extensions of cyclotomic fields, and there
the orders he worked in did not match the ring of integers. As
for quadratic fields, I vaguely recall that Kummer's theory of
ideal numbers could not easily be generalized from the cyclotomic
case to general number fields. In any case, Eisenstein generalized
it to abelian extensions of Q(i).
Moreover, Kummer *knew* that Z[\zeta_3] was Euclidean, and he even
gave a proof for Z[\zeta_5].
> Also: Presumably at some point someone realized the connection between
> quadratic fields and Gauss's theory of quadratic forms, so that Gauss'
> computation of class numbers could be represented with hind-sight as
> giving proofs of unique factorization in those cases where the class
> number turned out to be 1.
That's what Dedekind did in his supplements to Dirichlet's lectures.
But, as I said, unique factorization in R = Z[(1+sqrt(-3))/2] was proved
by Gauss (unpublished till after his death) as well as Eisenstein
(also unpublished; he explicitly says at the beginning of
[Beweis des Reciprocit\"atssatzes f\"ur die cubischen Reste,
Crelle 27 (1844), 289--310] that he considers this to be well
known since everything is completely analogous to what Gauss had
done for Z[i]); Eisenstein even proved that Z[\sqrt{-1},\sqrt{2}] is
Euclidean in his big paper on the division of the lemniscate).
franz