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ACTA MATHEMATICA UNIVERSITATIS COMENIANAE **

p. 273 - 278

R. A. Mollin

**Abstract.** We consider the Diophantine equation of the form x^{2} – Dy^{2} = c, where c | 2D,
gcd(x,y) = 1, and provide criteria for solutions in terms of congruence conditions on the
fundamental solution of the Pell Equation x^{2} – Dy^{2} = 1. The proofs are elementary,
using only basic properties of simple continued fractions. The results generalize various
criteria for such solutions, and expose the central norm, defined by the infrastructure of
the underlying real quadratic field, as the foundational key that binds all the
elements.

**Keywords**:
Quadratic Diophantine equations, continued fractions, central
norms, fundamental unit.
**AMS Subject classification:** Primary: 11D09, 11R11, 11A55;
Secondary: 11R29.

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