Given coprime positive integers d′,d′′, B\'ezout's Lemma tells us that
there are integers u,v so that d′u−d′′v=1. We show that, interchanging d′
and d′′ if necessary, we may choose u and v to be Loeschian numbers,
i.e., of the form ∣α∣2, where α∈Z[j], the ring of
integers of the number field Q(j), where j2+j+1=0. We do this by
using Atkin-Lehner elements in some quaternion algebras H. We use
this fact to count the number of conjugacy classes of elements of order 3 in an
order O⊂H.Comment: 20 pages, comments welcom