  We establish Schmutz Schaller's conjecture that the hexagonal lattice is
`better' than the square lattice.
  Schmutz Schaller (Bulletin of the AMS 35 (1998), p. 201), motivated by
considerations from hyperbolic geometry, conjectured that in dimensions 2 to 8
the best known lattice sphere packings have `maximal lengths' and goes on to
write: "In dimension 2 the conjecture means in particular that the hexagonal
lattice is `better' than the square lattice. More precisely, let 0<h_1<h_2<...
be the positive integers, listed in ascending order, which can be written as
h_i=x^2+3y^2 for integers x and y. Let 0<q_1<q_2<... be the positive integers,
listed in ascending order, which can be written as q_i=x^2+y^2 for integers x
and y. Then the conjecture is that q_i<=h_i for i=1,2,3,..."
  Our proof requires computational prime number theory in combination with
methods from a preprint of the first author (to appear in Math. Comp.),
arXiv:math.NT/0112100.
