State/Periodicity along the river

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Proposition
Proof

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

State/Periodicity along the river

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	Suppose that $Q$ is a binary quadratic form of positive, nonsquare discriminant $\Delta$.
	Choose a river segment, in the river of the range topograph of $Q$. Let $p_0$ be the positive integer adjacent to the river segment, and $n_0$ the negative integer adjacent to the river segment. In addition, choose a direction to travel along the river, and let $p_i$, $n_i$ be the positive and negative integers adjacent to the river segment obtained by travelling $i$ units. Similarly, let $p_{-i}, n_{-i}$ denote the integers adjacent to the river segment obtained by travelling $i$ units in the opposite direction. Since every arc of the river has finite length, we find that the river contains an infinite number of riverbends. Since there are only finitely many riverbends of discriminant $\Delta$, some riverbend must occur twice along the river. It follows that there exist two integers $a,b$, such that: $p_a = p_a$. $n_a = n_b$. There is a river bend at $i$, and at $j$. The river bends at $i$ and at $j$ are identical. Let $m = a-b$. Since any cell along the river determines the following cells along the river, we find that: $p_i = p_{i+m}$, for all $i \in \ZZ$. $n_j = n_{j+m}$ for all $j \in \ZZ$.
	Hence the numbers along the river are periodic.