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# State/Solving quadratic Diophantine equations in two variables

Proposition: (Solving quadratic Diophantine equations in two variables) Suppose that $Q(x,y) = ax^2 + bxy + cy^2$ is an integer-valued binary quadratic form. Suppose that $n \in \ZZ$. Let $S$ be the following set: $$S = \{ (x,y) \in \ZZ \mbox{ such that } Q(x,y) = n \}.$$

The cardinality of $S$ can be determined algorithmically in finite time.

## Logical Connections

This statement logically relies on the following definitions and statements: Def/Binary quadratic form, Def/Discriminant of a binary quadratic form, State/Existence of wells for definite forms, State/Climbing in topographs, Def/Perfect square, Def/Range topograph, State/Periodicity along the river

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