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# State/Periodicity along the river

Proposition: (Periodicity along the river) Suppose that $Q$ is a integer-valued binary quadratic form. Suppose that the discriminant of $Q$ is positive, and not a perfect square. Thus, the range topograph of $Q$ contains an endless river.

Consider the sequence $(\ldots p_{-2}, p_{-1}, p_0, p_1, p_2, \ldots)$ of positive integers adjacent to the river in the range topograph. Also, consider the sequence $(\ldots, n_{-2}, n_{-1}, n_0, n_1, n_2, \ldots)$ of negative integers adjacent to the river in the range topograph.

Then, the sequences $(p_i)$ and $(n_j)$ are periodic. In other words, there exists a positive integer $m$, such that:

• $p_i = p_{i+m}$ for all $i \in \ZZ$.
• $n_j = n_{j+m}$ for all $j \in \ZZ$.

## Logical Connections

This statement logically relies on the following definitions and statements: Def/Binary quadratic form, Def/Discriminant of a binary quadratic form, Def/Range topograph, State/Every arc of river is finite, State/Bounding riverbends of a given discriminant

The following statements and definitions rely on the material of this page: State/Solving quadratic Diophantine equations in two variables

To visualize the logical connections between this statements and other items of mathematical knowledge, you can visit the following cluster(s), and click the "Visualize" tab: Clust/Binary quadratic forms