  The coincidence problem for planar patterns with $N$-fold symmetry is
considered. For the N-fold symmetric module with $N<46$, all isometries of the
plane are classified that result in coincidences of finite index. This is done
by reformulating the problem in terms of algebraic number fields and using
prime factorization. The more complicated case $N \ge 46$ is briefly discussed
and N=46 is described explicitly.
  The results of the coincidence problem also solve the problem of colour
lattices in two dimensions and its natural generalization to colour modules.
