# Article

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Keywords:
quadratic form; indefinite form; cycle; proper cycle; right neighbor
Summary:
In this paper we consider proper cycles of indefinite integral quadratic forms $F=(a,b,c)$ with discriminant $\Delta$. We prove that the proper cycles of $F$ can be obtained using their consecutive right neighbors $R^i(F)$ for $i\ge 0$. 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 $\tau (F)=(-a,b,-c)$ and $\chi (F)=(-c,b,-a)$.
References:
[1] J.  Buchmann: Algorithms for Binary Quadratic Forms. Springer-Verlag, accepted. Zbl 0948.11051
[2] D.  E.  Flath: Introduction to Number Theory. Wiley, New York, 1989. MR 0972739 | Zbl 0651.10001

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