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 Δ. 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 l of the cycle of F is odd, using the transformations τ(F) = (−a, b, −c) and ϰ(F) = (−c, b, −a).