Proper cycles of indefinite quadratic forms and their right neighbors

Abstract

In this paper we consider proper cycles of indefinite integral quadratic forms F = (a, b, c) with discriminant A. We prove that the proper cycles of F can be obtained using their consecutive right neighbors R-i(F) for i >= 0. We also derive explicit relations in the cycle and proper cycle of F when the length I of the cycle of F is odd, using the transformations tau(F) = (-a, b, -c) and chi(F) = (-c, b, -a).