  Here we constructively classify quadratic $d$-numbers: algebraic integers in
quadratic number fields generating Galois-invariant ideals. We prove the subset
thereof maximal among their Galois conjugates in absolute value is discrete in
$\mathbb{R}$. Our classification provides a characterization of those real
quadratic fields containing a unit of norm -1 which is known to be equivalent
to the existence of solutions to the negative Pell equation. The notion of a
weakly quadratic fusion category is introduced whose Frobenius-Perron dimension
necessarily lies in this discrete set. Factorization, divisibility, and
boundedness results are proven for quadratic $d$-numbers allowing a systematic
study of weakly quadratic fusion categories which constitute essentially all
known examples of fusion categories having no known connection to classical
representation theory.
