Cyclic subgroups of ideal class groups in real quadratic orders

The primary purpose of this paper is to provide general sufficient conditions for any real quadratic order to have a cyclic subgroup of order n is an element of N in its ideal class group. This generalizes results in the literature, including some seminal classical works. This is done with a simpler approach via the interplay between the maximal order and the non-maximal orders, using the underlying infrastructure via the continued fraction algorithm. Numerous examples and a concluding criterion for non-trivial class numbers are also provided. The latter links class number one criteria with new prime-producing quadratic polynomials.